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3JEMENTARY   COURSE  IN 


GRAPHIC  MATHEMATICS 


BY 


MATILDA   AUERBACH 

INSTRUCTOR   IN    MATHEMATICS,    ETHICAL   CULTURE   HIGH   SCHOOL 
NEW   YORK   CITY 


A  LLYN   and    BACON 
Boston       Neto  |forfe       Chicago 


AN  ELEMENTARY  COURSE  IN 


GRAPHIC  MATHEMATICS 


BY 

MATILDA   AUERBACH 

INSTRUCTOR   IN   MATHEMATICS,   ETHICAL  CULTURE   HIGH    SCHOOL 
NEW   YORK  CITY 


ALLYN    and   BACON 
Boston  jfteto  pork  Cbtcap 


AS 


Copyright,  1910, 
By  MATILDA  AUERBACH. 


Norton  on  Press : 

Berwick  &  Smith  Co.,  Norwood,  Mass.,  U.S.A. 


PREFACE, 

The  object  of  this  little  book  is  threefold: — first,  to 
show  the  pupil  some  practical  uses  of  the  graphic  method; 
second,  to  plan  a  course  in  graphic  algebra  that  will  lead 
naturally  and  along  interesting  paths  to  the  work  in  the 
solution  of  equations;  and  finally  to  save  both  teacher 
and  pupil  time  and  energy  needed  to  hunt  up  suitable 
material. 

Every  type  of  work  outlined  in  the  book  has  been 
tested  and  found  suitable  for  classroom  use.  The  writer 
has  done  a  considerable  amount  of  work  in  this  line  with 
her  classes  for  the  past  nine  years,  and  has  never  failed 
to  find  it  a  spring  by  means  of  which  she  has  been 
enabled  to  arouse  an  interest  in  the  mathematics. 

Though  elementary  in  its  form,  it  is  believed  the 
monograph  will  be  found  to  be  thoroughly  scientific. 
It  endeavors  to  introduce  in  simple  form  ideas  which 
the  pupil  will  come  to  deal  with  in  more  advanced  work 
and  in  no  case  introduces  an  idea  which  must  sooner  or 
later  be  unlearned. 

In  the  Appendix  at  the  end  of  the  book  may  be  found 
a  number  of  statistical  tables,  obtained  chiefly  from  the 
Bureau  of  Statistics  at  Washington,  from  which  teacher 
and  pupil  may  freely  draw  without  waste  of  time.  The 
writer  has  aimed  to  cover  a  wide  variety  of  topics  and  at 
the  same  time  to  select  those  in  which  figures  were  not 
too  large  for  convenient  use. 

MATILDA  AUERBACH, 
Ethical  Culture  High  School. 

259726 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/elementarycourseOOauerrich 


CONTENTS 


CHAPTER  I 
Introductory:  The  Meaning  of  the  Graph    .         .       i 

CHAPTER  II 
Some  Practical  Uses  of  the  Graph  ...       6 

a.  IN  SURVEYING  .......         6 

b.  IN  KEEPING    STATISTICS,    RECORDS,  AND  AS  READY 

RECKONERS          8 

C.  IN  REPRESENTING  FORMULAS  .  .  .  •  r3 
d.    IN    THE    SOLUTION    OF    PROBLEMS    INVOLVING    THE 

ELEMENT  OF  TIME l8 


CHAPTER  III 
Study  of  the  Function  and  Equation 

a.  THE  FUNCTION  .  .  . 

b.  THE  EQUATION 

I.    SINGLE  LINEAR    . 
II.    SIMULTANEOUS  LINEAR 


III.  SINGLE  QUADRATIC  AND  THOSE  OF  HIGHER 

DEGREE  ..... 

IV.  SIMULTANEOUS  LINEAR  AND  QUADRATIC 
V.    SIMULTANEOUS  QUADRATIC 


22 

28 
28 
29 

31 

33 
34 


APPENDIX 
Appendix  to  Chapter  ii 35 


CHAPTER  I 

INTRODUCTORY:    THE.  MEANING  OF  A 
GRAPH 

We  all  have  had  the  experience  of  wishing  to  place  a 
point  somewhere  definitely  upon  a  sheet  of  paper,  upon 
the  blackboard,  or  upon  some  flat  surface.  How  have 
we  done  it  ?  What  have  we  really  done  when  we  have 
said  the  point  is  to  be  three  inches  from  the  lower  edge 
and  two  inches  from  the  right  edge?  We  have  done 
practically  what  we  do  when  we  say  New  York  City  is 
74°  West  longitude  and  41°  North  latitude.  We  have 
drawn  two  lines  (either  real  or  imaginary)  in  the  first 
case,  one  three  inches  above  the  lower  edge  and  the 
other  two  inches  to  the  left  of  the  right  edge  of  the 
paper,  and  have  found  the  point  at  their  crossing — in 
the  second  case  we  have  drawn  one  line  through  a  point 
on  the  equator  just  74°  to  the  left  of  the  meridian  through 
Greenwich,  and  another  line  parallel  to  the  equator 
just  41°  above  it.  Their  point  of  intersection  has  again 
given  us  the  desired  point.  In  the  same  manner  we 
could  construct  any  map — one  of  the  city,  showing  points 
of  interest — one  of  a  piece  of  ground  that  has  been 
surveyed,  or  anything  of  the  sort,  just  by  referring  each 
of  the  points  in  question  to  two  intersecting  lines.  These 
lines  are  known  as  axes,  and  in  all  elementary  work  are 
drawn  at  right  angles  to  each  other. 

1 


2  GKA  PHIC  MA  THEM  A  TICS 

EXERCISES 

1.  If  West  longitude  is  reckoned  to  the  left  of  the 
Greenwich  axis,  how  will  East  longitude  be  reckoned? 
If  North  latitude  is  reckoned  up  from  the  equator,  how 
will  South  latitude  be  reckoned  ? 

2.  Using  the  Greenwich  meridian  and  equator  as 
axes,  locate  the  following  cities: 

(1)  New  York  (74°  W.,  41°  N.) 

(2)  St.  Petersburg  (30°  E.,  60°  N.) 

(3)  Buenos  Ayres  (58°  W.,  35°  S.) 

(4)  San  Francisco  (122°  W.,  37°  N.) 

(5)  Zanzibar  (49°  E.,  6°  S.) 

(6)  London  (0°,  51£°  N.) 

3.  Using  any  two  streets  that  run  at  right  angles  to 
each  other  as  axes,  locate  at  least  a  dozen  points  of 
interest  in  the  city  in  which  you  live. 

In  locating  points  in  general  with  respect  to  two 
axes,  matters  may  be  greatly  simplified  by  using  positive 
and  negative  numbers. 

EXERCISES 

1.  List  the  following  words  and  phrases  under  the 
two  heads  " positive"  and  "negative": — right,  wrong; 
debit,  credit;  right,  left;  below,  above;  above  zero, 
below  zero;  B.  C,  A.  D.;  East,  West,  North,  South;  sane, 
insane;  pauper,  tax-payer;  time  to  come,  time  past; 
increase  in  population,  decrease  in  population. 

2.  Which  of  the  above  might  be  considered  as  lying 
to  the  right  of  a  vertical  axis?  Which  to  the  left? 
Which  above  a  horizontal  axis  ?     Which  below  it  ? 

We  have  seen  that  to  locate  a  point  on  a  plane  surface, 
reference  must  be  made  to  two  axes,  for  there  are  in- 
numerable points  that  lie  four  inches  to  the  right  of  a 


THE  MEANING  OF  A   GRAPH  3 

vertical  axis,  while  there  is  but  one  that  lies  at  the  same 
time  5  inches  below  a  horizontal  axis. 

EXERCISES 

Suppose  we  take  the  turning  point  from  the  year  1907 
to  the  year  1908  as  our  zero  point  on  the  horizontal  axis 
in  this  diagram,  (Fig.  1),  and  the  temperature  0°  Fahren- 


~W 

_          _                 .£- 

& 

_      __          _          X   |Jt 

±~ 

-v- 

_iTim>Axis 



Fig.l 

heit  as  our  zero  point  on   the   vertical  axis: — 

1.  Where  will  all  points  representing  time  previous  to 
Jan.  1,  1908,  be  located?  Where  all  those  representing 
time  after  that  date?  Where  all  those  representing 
temperature  below  zero?  and  where  all  those  repre- 
senting temperature  above  zero  ? 

2.  Through  what  point  would  you  draw  an  imaginary 
line  to  represent  mid-day,  Jan.  5,  1908,  if  each  day  of 


4  GRAPHIC  MA  THEM  A  TICS 

24  hours  is   represented   by   12  small   divisions   on   the 
diagram?     Dec.  25,  1907,  6  p.m.?     Jan.  10,  1908,  8  a.m.? 

3.  Through   what   point   would   you   draw   a   line   to 
represent  the  temperature  5°  above  zero  (that  is,  +5°)? 
7°  below  zero?     12°  below  zero? 

4.  Look  up  the  temperature  for  each  day  of  the  past 
week,  and  record  it  by  means  of  a  diagram. 

For  more  complicated  problems  of  this  type  see  Ap- 
pendix to  Chapter  II. 

As  you  may  already  have  observed,  we  can  in  general 
locate  points  in  the  four  quadrants  into  which  our  surface 
is  divided  by  the  two  axes  in  the  following  manner. 
Suppose  the  distance  of  all  points  to  the  right  or  left  of 
the  vertical  or  yy'  axis  in  the  diagram  (Fig.  2)  be  denoted 


~    Ti 

i)~ 

v°~ 

",3J 

-J 

YT 

Fig.2 

by  x,  and  the  distance  of  all  those  above  or  below  the 
the  horizontal  or  xx'  axis  be  denoted  by  y.  Then  when 
x  is  positive  the  distance  is  measured  so  many  units  to 


THE  MEANING  OF  A   GRAPH  5 

the  right,  and  when  it  is  negative,  so  many  units  to  the 
left  of  the  yy'  axis.  When  y  is  positive  the  distance  is 
measured  so  many  units  above  the  xx'  axis,  and  when 
negative  so  many  below  it.  For  instance,  suppose  the 
the  point  (x,  y)  =  (7,  12)  be  given.  It  will  be  in  the  first 
quadrant,  (1,  Fig.  2),  on  an  imaginary  line  7  units  to  the 
right  of  yy/  and  parallel  to  it,  and  on  another  such  line 
12  units  above  xx/  and  parallel  to  it — namely  point  P. 
If  a  point  is  described  as  (xy  y)  =  (—7,  12)  it  will  lie  in 
quadrant  II,  7  units  across  to  the  left,  and  12  units  up, 
namely  point  Px.  {x,  y)  =  (—7,  —12)  will  lie  in  the  third 
quadrant  7  units  across  to  the  left,  and  12  down,  point  P9t 
and  finally  the  point  (x,  y)  =  (7,  —  12)  lies  in  quadrant  IV, 
7  units  across  to  the  right,  and  12  units  down,  point  P3. 

EXERCISES 

1 .  Locate  the  points  (9, 11),  (7,  6),  (—15, 17),  (—19,  —20), 
(-2,  6),  (8,  -15),  (7,  -13),  (-11,  -9),  (-2,  15). 

2.  Locate  the  points  (1,  5),  (3,  7),  (5, 2),  (9,  —3),  (12,  —6) 
and  draw  a  line  connecting  them. 

Any  line  (curved,  broken  or  straight)  drawn  through  a 
series  of  fixed  points  as  in  the  last  exercise  is  called  a 
graph. 

EXERCISE 

1.  Draw  the  graph  determined  by  the  points  ( — 3,  — 2), 
(-1,  0),  (0,  1),  (2,  i),  (5,  7),  (8,  -11). 


CHAPTER  II 

50ML  OF  THL   PRACTICAL   U5E.5  OF  THL 

GRAPH 

Now  that  we  have  learned  to  locate  points  in  this 
simple  manner,  we  are  ready  for  a  few  simple  practical 
applications  in  addition  to  the  above. 

IN  SURVEYING 
EXERCISES 

1.  In  surveying-  a  hexagonal  field  a  surveyor  notes 
the  following  points  as  its  vertices:  A  =  (6,  7),  B  =  (20,  20), 
C=(40,'20),  Z>=  (35,  0),E=  (10,-20)  andi^  =  (0,-10). 
Plot  the  points,  and  draw  the  outline  of  the  field.  Find 
the  number  of  square  units  in  the  area  of  the  field  in 
two  ways: —  (1)  By  breaking  the  diagram  of  the  field 
into  figures  of  which  you  can  find  the  areas  and  adding 
them,  (2)  By  a  process  of  subtraction,  using  the  square 
whose  vertices  are  denoted  by  the  points  (0,  20),  (40,  20), 
(40,  —20),  (0,  —20). 

2.  It  is  customary  among  surveyors  to  have  the 
polygon  lie  eventually  entirely  in  the  first  quadrant. 
Can  you  see  any  reason  for  this  ? 

3.  Through  how  many  units  will  you  have  to  move 
the  polygon  indicated  in  Ex.  1,  so  that  it  shall  just  lie 
wholly  in  the  first  quadrant  ? 

4.  Will  all  the  values  indicating  the  vertices  be 
changed? 

5.  Describe  the  new  positions  of  A,  By  C\  D,  £y  and  F. 

6 


PRACTICAL   USES  OF  THE  GRAPH  7 

6.  The  vertices  of  a  pentagonal  field  are  located  by 
the  following  points,  A  =  (—20,  15),  B  =  (10,  20),  C  = 
(23,  —20),  D  =  (—10,  —30),  E  =  (—30,  —10). 

(1)  Draw  the  outline  of  the  field. 

(2)  Give  new  values  to  A,  £,  C,  Dy  E,  so  that  the 
area  shall  remain  the  same  but  the  diagram  lie 
wholly  in  the  first  quadrant  with  E  on  the  North- 
South  axis,  and  D  on  the  East- West  axis. 

(3)  Find  the  area  of  the  field. 


7.  From  the  accompany- 
ing diagram  (Fig.  3),  find 
the  approximate  area  of  the 
pond. 


Eig.3 


8.  The  accompany- 
ing diagram  (Fig.  4), 
represents  the  survey 
of  a  field  with  curved 
boundary.  Find  the  ap- 
proximate area  of  the 
field. 


,'^L 

^l9 

^     ^%r 

_                                             __     _yef 

*     i- v^  it 

j —  \-^ 

~t"At 

X      s,st 

^JJ<! 

it       \5 

i        >5 

Tt-t'. 

""^p9 

si-  - 

Jit 

t  + 

7  + 

t  Ij 

-,22- 

£Z-  - 

^hT"  1  s  *'" 

'El. 

1  r 

Fig.4 


GRAPHIC  MA  THEM  A  TICS 


IN   KEEPING  STATISTICS  AND  AS   READY 
RECKONERS 

9.  The  following  table  gives  the  highest  and  lowest 
prices  in  New  York,  for  Middling  Uplands  Cotton  from 
Jan.  1  to  Dec.  31  of  the  years  named.  Show  the  graph 
of  the  highest  in  red  ink  and  that  of  the  lowest  in  black 
ink  on  the  same  pair  of  axes,  and  correct  to  the  nearest 
half. 


YEAR 

HIGHEST 

LOWEST 

YEAR 

1864 

HIGHEST 

LOWEST 

YEAR 

1872 

HIGHEST 

LOWEST 

1826 

14 

9 

190 

72 

m 

18| 

1835 

25 

15 

1865 

120 

35 

1873 

m 

13| 

1840 

10 

8 

1866 

52 

32 

1874 

18* 

14| 

1850 

14 

11 

1867 

36 

15* 

1885 

13± 

mi 

1860 

11* 

10 

1868 

33 

16 

1890 

12| 

»A 

1861 

38 

Hi 

1869 

35 

25 

1895 

9ft 

h\ 

1862 

69^ 

20 

1870 

25f 

15 

1863 

93 

51 

1871 

21$ 

14| 

10.  What  facts  does  the  graph  of  the  table  in  Ex.  9 
bring  out  clearly  before  you? 

11.  Calling  one  the  time  axis,  and  the  other  the  popu- 
lation axis,  draw  graphs  indicating  the  following  sets  of 
data: 

(1)  The  population  of  the  United  States  per 
square  mile: 

YEAR  POP.  YEAR  POP. 

1800 6.41     1900 25.22 

1850 7.78     1904 27.02 

1870 12.74 

(2)  The  population  of  England,  Ireland,  Scotland, 
and  Wales  correct  to  the  nearest  10,000:  (Draw  the 
graphs  using  a  single  pair  of  axes,  a  different  kind 
of  line  for  each,  and  correct  to  the  nearest  100,000.) 


PRACTICAL   USES  OF  THE  GRAPH 


YEAR 

ENGLAND 

IRELAND 

SCOTLAND 

WALES 

1831 

13,090,000 

7,770,000 

2,360,000 

810,000 

1841 

15,000,000 

8,200,000 

2,620,000 

910,000 

1851 

16,920,000 

6,570,000 

2,890,000 

1,010,000 

1861 

18,950,000 

5,800.000 

3,060,000 

1,110,000 

1871 

21,500,000 

5,410,000 

3,360,000 

1,220,000 

1881 

24,610,000 

5,180,000 

3,740,000 

1,360,000 

1891 

27,500,000 

4,710,000 

4,030,000 

1,500,000 

1901 

32,530,000 

4,460,000 

4,470,000 

* 

*  After  1891  merged  into  England. 

12.  Answer  the  following  questions  from  the  graphs 
drawn  in  Ex.  11,  (2): 

(1)  In  approximately  what  year  was  the  popu- 
lation of  England  17  million  ? 

(2)  What  was  the  population  of  England  in  1835? 
in  1845?     in  1865?     in  1875? 

(3)  In  which  of  the  four  countries  has  the  popu- 
lation increased  least  rapidly  ?     Most  rapidly  ? 

(4)  In  which  has  there  been  a  decrease  ? 

(5)  In  what  year  was  the  population  of  two  of 
them  practically  the  same  ?  In  which  countries  was 
this  the  case  ? 

(6)  Roughly  speaking,  when  will  the  population 
of  England  be  38  million?  (/.  e.  considering  the 
increase  to  continue  uniformly.) 

(7)  What  will  be  the  population  of  each  of  the 
others  at  that  time  ? 

(8)  When  will  that  of  Ireland  and  Wales  be  the 
same  ?    What  will  it  be  at  that  time  ? 

(9)  Will  this  happen  apparently  in  the  case  of 
Scotland  and  Wales? 

For  other  problems  of  this  type  see  Appendix  to 
Chapter  II. 

The  graphic  method  of  recording  the  readings  of  a 
thermometer  and  barometer  has  been  adopted  by  many 
newspapers. 


10 


GRAPHIC  MA  THEM  A  TICS 


EXERCISES 

1.  Observe  the  readings  of  the  same  thermometer  at 
the  same  hours  daily  for  a  week,  and  record  the  results 
of  your  observations  graphically. 

2.  Record  graphically  the  readings  of  the  barometer 
as  taken  from  the  same  newspaper  daily  for  a  week. 

3.  Record  graphically  the  scores  of  the  captains  of  the 
girls'  and  boys'  basket  ball  teams  in  your  school.  (One 
in  red  and  the  other  in  black  ink,  or  one  by  means  of  a 
solid  and  the  other  by  means  of  a  dotted  line.) 

4.  The  Harvard  Eights  from  1852  through  1905  had 
rowed  39  races.     The  records  are  as  follows: 


TIME 

TIME 

DATE 

WON  BY 

WINNER 

LOSER 

DATE 

WON  BY 

WINNER 

LOSER 

1852 

Harvard 

1884 

Yale 

20  31 

20.46 

1855 

a 

1885 

Harvard 

25.15 

26.30 

1857 

ll 

19.18 

20.18 

1886 

Yale 

20.41 

21.05 

1859 

Yale 

19.14 

19.16 

1887 

i< 

22.56 

23.11 

1860 

Harvard 

18.53 

19.05 

1888 

u 

20.10 

21.24 

1864 

a 

19.01 

19.43 

1889 

II 

21.30 

21.55 

1865 

Yale 

17.42 

18.09 

1890 

« 

21.29 

21.40 

1866 

Harvard 

18.43 

19.10 

1891 

Harvard 

21.23 

21.57 

1867 

a 

18.13 

19.25 

1892 

Yale 

20.48 

21.42 

1868 

a 

17.48 

18.30 

1893 

u 

25.01 

25.15 

1869 

a 

18.02 

18.11 

1894 

a 

22.47 

24.40 

1870 

a 

Foul 

Disq. 

1895 

ci 

21.30 

22.05 

1876 

Yale 

22.02 

22.33 

1899 

Harvard 

20.52 

21.13 

1877 

Harvard 

24.36 

24.44 

1900 

Yale 

21.13 

21.37 

1878 

t« 

20.45 

21.29 

1901 

it 

23.37 

23.45 

1879 

u 

22.15 

23.58 

1902 

u 

20.20 

20.33 

1880 

Yale 

24.27 

25.09 

1903 

(I 

20.20 

20.30 

1881 

u 

22.13 

22.19 

1904 

II 

21.40 

22.10 

1882 

Harvard 

20.47 

20.50 

1905 

II 

22.33 

22.36 

1883 

a 

24.26 

25.59 

Show  this  graphically. 


PRACTICAL  USES  OF  THE  GRAPH  11 

As  seen  above  in  plotting  population  curves,  valuable 
surmises  might  be  made  in  regard  to  probable  increase 
or  decrease  in  populations  during  specified  periods,  or 
rough  estimates  could  be  made  as  to  the  probable  popu- 
lations at  any  stated  time,  and  so  forth.  Likewise,  there 
is  another  use  of  the  graph  in  the  way  of  a  "ready- 
reckoner"  where  price  lists  do  not  include,  for  instance, 
all  sizes  of  articles  or  numbers  of  articles  of  the  same 
kind  for  sale.  This  will  be  made  clear  by  the  following 
set  of  problems: 

1.  The  single  ticket  by  railway  costs  $2.50.  If  10  such 
tickets  be  purchased  the  average  cost  will  be  reduced  to 
$2.25.  If  50  be  purchased  the  cost  per  ticket  will  be 
only  $1.80;  if  100,  the  cost  per  ticket  will  be  $1.50;  and  if 
200,  the  cost  per  ticket  will  be  $1.25.  Draw  a  graph 
showing  this,  and  answer  the  following  questions  by  the 
aid  of  it: 

(i)  What  will  be  the  probable  cost  per  ticket  if 
an  excursion  of  75  be  formed?  If  one  of  125  be 
formed?     One  of  175? 

(2)  About  how  many  tickets  must  be  used  to 
reduce  the  expense  per  head  to  just  $2.00  ?  to  $1.60  ? 

2.  If  a  certain  kind  of  desk  be  sold  to  the  individual 
it  will  cost  $30.00.  If  ordered  by  the  dozen  it  will  cost 
$28.50,  if  6  dozen  are  ordered  it  will  cost  $22.50,  and  if 
150  are  ordered  the  cost  will  fall  as  low  as  $20.00.  Draw 
a  graph  showing  this,  and  answer  the  following  questions: 

(1)  What  will  be  the  probable  cost  per  desk 
when  36  are  ordered?     When  100  are  ordered? 

(2)  How  many  must  be  ordered  so  that  each 
shall  cost  about  $25.00? 

3.  Ordering  ink  by  the  gill  it  costs  $  .10.  By  the 
pint  it  costs  $  .30,  by  the  quart  $  .50,  and  by  the  gallon 


12 


GRAPHIC  MA  THEM  A  TICS 


$1.75.  According  to  this,  what  should  it  cost  approxi- 
mately when  ordered  by  the  half-gallon?  By  the  half- 
pint?     By  the  quart  and  a  pint? 

4.  The  average  annual  premiums  (P)  for  whole  life 
insurance  of  $500  for  the  age  (A)  at  entry  is  given  as 
follows: 


A  = 

21 

25 

30 

35 

40 

45 

50 

P  = 

- 

$8.00 

$8.66 

$10.00 

$11.66 

$14.00 

$16  75  $20.10 

What  are  the  probable  premiums  for  ages  23,  27,  33, 
37,42,  48? 

5.  It  is  found  by  testing,  that  the  barometer  stands  at 
30  inches  at  sea  level,  at  23.5  inches  at  a  height  of  6,000 
feet,  at  18.2  inches  at  a  height  of  12,000  feet,  at  12.2 
inches  at  24,000  feet,  and  at  7.3  inches  at  36,000  feet  above 
sea  level.  Plot  the  graph  indicating  these  facts,  and 
from  it  answer  the  following  questions: — 

(1)  How  high  (approximately)  is  a  place  in  which 
the  barometer  stands  at  25  inches?     At  20  inches ? 

(2)  How  high  should  the  barometer  rise  in  a 
spot  which  is  20,000  feet  above  sea  level  ?  At  one 
which  is  30,000  feet  above  sea  level  ? 

6.  In  a  price  list  the  following  table  appears: 


Measuring-tins  of 
capacity  P  (pints)  = 

1 

2 

3 

4 

6 

8 

12 

Cost  in  cents  C  = 

10 

16 

21 

24 

30 

35 

42 

What  will  tins  of  a  capacity  of  5  pints,  7  pints,  9,  10,  11 
pints  respectively,  probably  cost  ? 


PRACTICAL  USES  OF  THE  GRAPH 


13 


7.     The  cost  of  fitted   lunch  baskets   is  given   in  the 
following  table: 


Arranged  for  number 
of  persons  N  = 

1 

2 

4 

6 

Cost  in  dollars  D  = 

10 

18 

30 

40 

What  will  be  the  probable  cost  of  baskets  for  3,  5,  7,  8, 
and  10  persons  respectively  ? 


IN  REPRESENTING  FORMULAS 

In  the  last  set  of  applications  of  the  graph  we  have 
seen  that  by  joining  successive  given  points  by  straight 
lines,  we  may  surmise  approximate  results  for  inter- 
mediate points.  However,  •  there  has  been  no  law 
governing  the  statements  thus  made,  and  the  results 
obtained  may  or  may  not  satisfy  existing  conditions.  In 
short,  it  was  only  a  surmise  on  our  part  when  we  drew 
conclusions. 

There  is,  however,  another  type  of  problem  which  may 
be  represented  or  approximately  solved  graphically — 
namely  those  which  rest  upon  a  formula.  For  instance, 
we  are  told  that  the  circumference  of  a  circular  is  always 
equal  to  it  times  its  diameter,  or  approximately  3^  times 
its  diameter.  That  is,  if  C  stands  for  the  number  of 
units  in  a  circumference,  and  D  for  the  number  of  units 
in  its  diameter,  C  =  7T  D. 

EXERCISES 
1.     Given  C  =  tyD,  where  C  =  number  units   in   the 
circumference  of  a  circle  and  D  =  number  units  in  its 
diameter: — 


14 


GRAPHIC  MA  THE  MA  TICS 


(1)  Find  the  values  of  C  for  those  given  in  the 
following  table  for  D. 


z>  = 

i 

14 

3* 

21 

28 

c  = 

(2)  Call  one  axis  {DDf),  the  diameter  axis,  and 
the  other   {CO),  the  axis  of  circumferences,  and 

)      plot  the  points  corresponding  to  the  values  found 
in  Ex.  (1). 

(3)  Connect  these  points  and  state  on  what  kind 
of  line  they  lie. 

(4)  How  many  of  these  points  would  have  been 
needed  to  enable  you  to  draw  that  line  ? 

(5)  From  the  line  you  have  drawn  find  answers 
to  the  following  questions: 

{a)  When  the  diameter  of  a  circle  is  10  units 
how  many  units  are  contained  in  its  circum- 
ference ? 

(b)  When  D  =  10J  ft.,  C  =? 

(c)  When  C  =  100,       D  =? 

(d)  When  C  =  75,        D  =? 

(e)  If  the  circumference  of  a  wheel  is  92 
inches,  what  is  the  length  of  its  diameter? 

2.     We  are  told  that  an  inch  contains  2.54  centimeters. 
Answer  the  following: 

(1)  The  number  of  centimeters  in  a  given  length 
is  then  always  how  many  times  the  number  of 
inches  in  that  length  ? 

(2)  Write  a  formula  stating  this  fact. 

(3)  As  in  Ex.  1  (1),  select  any  six  lengths  in 
terms  of  inches  and   make    a   table   showing   the 


PRACTICAL  USES  OF  THE  GRAPH  15 

number    of    centimeters     in     the     corresponding 
lengths. 

(4)  Call  one  axis  {IP),  and  the  other  (CO),  and 
plot  the  points  corresponding  to  the  values  found 
in  (3). 

(5)  On  what  kind  of  line  do  these  points  lie? 
Draw  it. 

(6)  How  many  of  these  points  would  have  been 
needed  to  enable  you  to  draw  that  line  ? 

(7)  From  the  graph  just  plotted,  answer  the 
following  questions: 

(a)  About  how  many  inches  in  30  cm.  ? 

(b)  About  how  many  centimeters  in  20  in.  ? 

(c)  About  how  many  inches  in  40  cm.? 

(d)  About  how  many  inches  in  a  meter  ? 

3.  The  formula  for  the  reduction  of  Fahrenheit  scale  to 
Centigrade  scale  is  C  =  |  (F  —  32)  where  C  =  the  num- 
ber of  degrees  Centigrade  corresponding  to  F  =  any 
given  number  of  degrees  Fahrenheit. 

(1)  Give  six  values  to  F,  and  as.  in  Ex.  1  (1),  show 
in  a  table  the  corresponding  values  of  C. 

(2)  Call  the  axes  of  Fahrenheit  and  Centigrade 
FFf  and  CO  respectively,  and  plot  the  points 
shown  in  this  table. 

(3)  Connect  these  points  and  tell  on  what  kind 
of  line  they  lie. 

(4)  How  many  of  these  points  would  have  been 
needed  to  enable  you  to  draw  that  line? 

(5)  From  the  lines  you  have  drawn  find  the 
approximate  number  of  degrees  on  a  Fahrenheit 
thermometer  when  a  Centigrade  thermometer 
registers  (a),  10°,  (b),  100°,  (c),  50°,  (d)}  120°,  (<?),  0°. 


1 6  GRAPHIC  MA  THE  MA  TICS 

(6)  From   the   same   line  find  the  approximate 
number  of  degrees  on  a  Centigrade  thermometer 
when  a  Fahrenheit  thermometer  registers  (a),  10°, 
(b),  20°,  (0,  35°,  (d),  180°,  (<?),  212°. 
4.     On  an  examination  paper  125  points  may  be  ob- 
tained. 

(1)  Write  a  formula  stating  this  fact  and  draw 
its  graph  as  in  the  above  exercises  so  that  the 
examiner  may  use  it  to  mark  the  set  of  papers. 
(That  is,  so  that  he  may  reduce  any  number  of 
points  to  per  cent.) 

(2)  What  per  cent,  will  pupils  have  who  have 
90  points,  10  points,  60  points,  115  points,  120  points 
correct? 

GENERAL  QUESTIONS 

1.  In  each  case  in  the  above  four  exercises,  the  formula 
was  of  what  degree  ? 

2.  In  each  case  what  was  the  result  in  plotting  the 
graph  of  the  formula  ? 

3.  In  each  case  how  many  points  were  needed  to  plot 
the  graph  of  the  formula  ? 

4.  Can  you  formulate  a  general  rule  as  to  advisability 
in  the  selection  of  these  points  ? 

It  has  been  possible  to  represent  each  of  the  foregoing 
formulas  by  means  of  a  straight  line.  There  are,  how- 
ever, many  that  cannot  be  so  represented.  The. following 
problems  will  make  this  point  clear. 

EXERCISES 

1.  The  area  of  a  circle  in  terms  of  its  radius  is  ex- 
pressed by  the  formula  A  =  it  K*.  Find  the  values  of  A 
when  R  =  1;  1|  2£,  3|,  4|,  7,  and  plot  the  corresponding 
points.     (Let  tt  =  3^,  call  the  axes  AA/  and  RR',  and  use 


PRACTICAL  USES  OF  THE  GRAPH 


17 


a  convenient  scale.)  Will  the  line  drawn  through  these 
points  be  a  straight  line?  Could  it  have  been  found 
from  any  two  of  the  points  used  ?  What  would  you 
have  to  do  to  find  a  more  accurate  graph  than  the  one 
you  have  found? 

2.     From  the  graph  drawn  in  Ex.  1,  answer  the  following 
questions: — 

(1)  What  is  the  approximate  area  of  the  circle 
whose  radius  is  3,  4,  5  feet  respectively  ? 

(2)  What  is  the  approximate  length  of  the 
radius  of  a  circle  when  its  area  is  150,  38  square 
units  respectively  ? 

When  a  body  falls  freely  from  rest,  the  space  in 
sy  through  which  it  travels  in  a  given  time  in 
seconds,  t,  is  expressed  by  the  formula  s  =  16  f.  What 
will  be  a  good  scale  to  use  in  plotting  the  graph  of  this 
formula?     Find  the  corresponding  values  of  s  when 


3. 
feet 


t  = 

0 

\ 

\ 

f 

1 

f 

1 

s  = 

Plot  the  graph  of  the  points  thus  found,  using  the  scale 
decided  upon. 

4.  From  the  graph  drawn  for  Ex.  3,  what  is  the  approxi- 
mate distance  through  which  a  body  falls  in  5,  2£,  3£, 
seconds  respectively? 

5.  From  the  same  graph  find  the  approximate  time 
needed  for  a  body  to  fall  64  ft.,  144  ft.,  120  ft. 

6.  About  how  high  is  a  building  if  a  ball  dropped  from 
the  roof  takes  3  seconds  to  reach  the  ground  ? 

7.  If  squares  of  brass  are  cut  from  a  sheet  of  uniform 
thickness,  their  weights  are  proportional  to  the  squares  of 


1 8  GRAPHIC  MA  THEM  A  TICS 

the  lengths  of  their  sides.  Write  a  formula  stating  this 
fact,  letting  u  stand  for  the  weight  of  a  unit  square, 
s  stand  for  the  length  of  a  side  of  any  square,  and  w  for 
the  weight  of  that  square. 

8.  Let  the  unit  square  weigh  -J  pound  and  plot  the 
graph  of  the  formula  obtained  in  Ex.  7. 

9.  From  the  graph  in  Ex.  8  find  the  approximate 
weights  of  squares  of  brass  whose  sides  are  2,  4,  5  units 
respectively. 

10.  Write  a  formula  and  from  it  construct  a  "ready- 
reckoner"  showing  the  price  of  pig-iron  at  $21.50  per  ton. 

11.  Construct  a  ready-reckoner  showing  that  a  litre 
equals  about  1.75  pints.  How  many  pints,  according  to 
this  graph,  in  2 J,  3 J,  4  litres,  respectively? 

12.  Construct  y  =  ;ra,  and  determine  from  the  graph 
i/2,  i/3,  i/5,  i/7,  i/8,  i/10,  -/IT,  i/T5^  approximately. 

13.  Construct  y  =  10x,  and  determine  the  values  of 
y  when  x  =  1.5,  — 1,  1.9,  —1.5,  2.5. 

IN  THE  SOLUTION   OF  PROBLEMS 
INVOLVING    THE    ELEMENT    OF    TIME 

Many  of  the  problems  involving  the  element  of  time 
may  be  solved  graphically.  Those  who  have  solved  a 
sufficient  number  of  the  foregoing  problems  will  need 
no  further  explanation  to  enable  them  to  answer  the 
following  questions: — 

EXERCISES 

1.  Call  the  shorter  axis  the  time  axis  (TT')  and  the 
longer  the  rate  axis  (RR'). 

Plot  the  ready-reckoner  showing  the  ground  covered 
by  a  man  whose  rate  is  3£  miles  per  hour.  (The  formula 
used  in  this  case  D  s  77?.)     Suppose  a  second  man,  who 


PRACTICAL  USES  OF  THE  GRAPH  19 

had  a  handicap  of  5  miles,  travels  at  the  rate  of  3  miles 
per  hour.  What  will  represent  his  starting  point  ?  Where 
will  he  be  at  the  end  of  three  hours  ?  At  what  point  do 
the  two  ready-reckoners  cross  each  other?  What  does 
this  point  tell  you? 

2.  A  steamboat  running  at  the  rate  of  8  miles  an  hour 
sees  a  motorboat  10  miles  off,  going  at  the  rate  of  5  miles 
per  hour.  How  far  will  the  steamboat  go  before  it 
overtakes  the  motorboat  ? 

3.  A  travels  6  miles  an  hour  and  B  8  miles  an  hour. 
If  A  starts  3  hours  before  B,  how  long  will  B  have  to 
travel  before  he  overtakes  A?  How  far  will  they  have 
travelled  before  this  occurs  ? 

4.  Two  cyclists,  A  and  B,  start  out  at  the  same  time. 
A  rides  for  1-J  hours  at  a  speed  of  10  miles  per  hour,  rests 
$  hour,  and  then  continues  on  his  course  at  7  miles  per 
hour.  B  rides  without  a  stop  at  the  rate  of  8  miles  per 
hour.     How  long  before  he  overtakes  A  ? 

5.  Two  men  start  at  the  same  time  to  walk  around  a 
circular  course  of  9  miles.  The  first  man's  rate  is  such 
that  he  completes  the  course  once  every  2£  hours,  and 
the  second  man's  such  that  he  completes  it  once  every  3 
hours.  How  long  after  starting  will  the  second  man 
pass  the  first?  How  long  before  he  will  pass  him  the 
second  time? 

(Hint:  At  what  point  will  a  man  be  when  he  has  gone 
the  course?  How  can  this  be  shown  using  simply  the 
pair  of  axes  and  no  curved  line  ?) 

6.  If  from  the  same  spot  on  a  circular  course  of  2 
miles  two  boys  walk  in  the  same  direction  at  the  rates  of 
5  and  3|  miles  an  hour  respectively,  how  often  and  at 
what  intervals  will  they  meet  if  they  continued  for  4 
hours?  If  they  walk  in  opposite  directions  how  often 
and  at  what  intervals  will  they  meet  ? 


20  GRAPHIC  MA  THEM  A  TICS 

7.  A  leaves  town  T  and  rows  at  the  rate  of  8-J-  miles 
per  hour  to  town  T'  and  back  again.  B  leaves  T'  at 
the  same  time  that  A  leaves  T,  and  rows  at  the  rate 
of  7  miles  per  hour  to  T.  Find  the  distance  between 
T  and  T*,  if  A  arives  at  town  T  3  hours  after  B. 

8.  A  train  meets  with  an  accident  after  travelling  1-J- 
hours.  The  accident  delays  it  2  hours,  after  which  it 
travels  at  £  its  former  rate,  and  arrives  at  its  destination 
2  hours  and  54  minutes  late.  If  the  accident  had  occurred 
48  miles  further  on,  the  delay  would  have  been  18  minutes 
less.  How  far  had  the  train  to  run,  and  what  were  its 
rates  before  and  after  the  accident  ? 

9.  A  man  rows  15  miles  up  a  river  and  back  again  in 
8  hours,  rowing  half  again  as  fast  with  the  stream  as 
against  it.  What  time  did  it  take  him  to  go  up  stream  ? 
What  were  his  rates  up  and  down? 

10.  Two  towns  T  and  T/  are  60  miles  apart.  A  walks 
from  T  to  T'  at  the  rate  of  3  miles  per  hour  and  trolleys 
back  at  the  rate  of  15  miles  per  hour.  B  starts  from 
T/  3  hours  later  than  A  from  T,  and  drives  to  T  at  the 
rate  of  6  miles  an  hour  and  walks  back  at  the  rate  of 
4  miles  an  hour.  How  long  after  starting  and  how  far 
from  T  do  they  meet  ? 

11.  In  how  many  years  will  the  interest  on  $600  equal 
the  amount  on  $200  if  both  are  invested  at  h%  ? 

12.  If  one  man  invests  $2,000  at  6%,  and  another  invests 
$10,000  at  5#,  in  how  many  years  will  the  amount  of  the 
first  man's  investment  equal  the  interest  on  the  second 
•man's. 

13.  In  how  many  years  will  the  interest  on  $500  at 
6$  differ  from  the  interest  on  $700  at  h%  by  $150  ? 

14.  Make  various  graphs  which  may  be  used  in  place 
of  "interest  tables." 


-200  - 


1 


~~1W: 


-20- 


hTnnt 


:fe: 


i 


'(2,64):: 


distance,  in  ieet ;  scale,  1 : 2.  Time,  in  seconds ;  scale,  16 : 1. 

Fig.  5.  —  Graph  of  the  Formula  s  =  16  tK 


CHAPTER  III 

STUDY  OF  THE  FUNCTION  AND  THE 
EQUATION 

The  work  we  had  in  the  preceding  chapter  in  the 
graphic  representation  of  formulas  will  help  us  to 
understand  the  following. 

In  the  first  place,  when  we  consider  the  formula  C~tt  d, 
we  see  at  once  that  whatever  value  we  give  to  d,  C  will 
have  a  corresponding  value.  That  is,  as  the  formula 
now  reads,  C  depends  for  its  value  upon  the  value  given 
to  d.  In  other  words,  the  values  of  d  and  C  may  vary  as 
much  as  we  please,  but  once  having  fixed  the  value  of  dy 
that  of  C  is  also  fixed.  In  this  case  both  d  and  C  are 
known  as  variable,  but  d  is  known  as  an  independent 
variable  and  C  as  a  dependent  variable.  If  we  were  to 
solve  the  equation  for  d  (that  is,  find  d  =  C  -^  it)  which 
would  be  the  dependent  and  which  the  independent 
variable  ?    Why  ? 

EXERCISES 

1.  Given  a  fixed  principal  and  a  fixed  rate  of  interest, 
upon  what  variable  would  the  amount  of  interest  depend  ? 

2.  Ordinarily,  upon  what  three  variables  does  the 
amount  of  interest  depend  ?  Write  a  formula  stating 
this. 

3.  Upon  what  two  variables  does  the  distance  a  man 
travels  depend?  Which  are  the  independent  and  which 
the  dependendent  variables  in  this  case  ? 

22 


THE  FUNCTION  AND  THE  EQUA  TION 


23 


4.  Give  illustrations  of  independent  and  dependent 
variables  in  life — in  nature. 

Every  dependent  variable  is  known  as  a  function  of 
the  independent  variable  or  variables  in  question.  For 
instance,  we  say  that  the  amount  of  interest  is  a  function 
of  the  independent  variables,  principal,  time,  and  rate. 
Likewise,  we  say  that  3  x*  +  5  x  +  6  is  a  function  of  x, 
for  it  depends  for  its  value  upon  the  value  given  the 
variable  x.  This  is  usually  written  f(x)  =  3  x*  +  5x  +  6. 
When  x  =  2,f(x)  becomes /(2)  =3  (2)2  +  5  (2)  +  6  =  28, 
and  it  is  readily  seen  that  as  we  give  different  values  to  x,, 
f(x)  will  have  correspondingly  different  values. 

Let  us  now  call  one  axis  the  ^r-axis,  and  the  other  (say 
the  vertical  axis),  the /"(.r)- axis,  and  attempt  to  plot  the 
graphs  of  f(x)  =  3  x  +  4  in  the  following  manner: — 

I.     Fill  in  the  values  omitted  in  the  table: 


Given  x  = 

— 5 

—2 

0 

2 

4 

then  3  x  = 
and 

f{x)  =  3  .r  +  4  = 

Thus  we  see  that  for  each  value   given  x,  we   have 
found  a  corresponding  value  forf(x). 

II.  Plot  the  points  representing  these  various  pairs  of 
values  of  x  and  f(x). 

III.  Draw  the, graph  determined  by  these  points,  and 
from  it  answer  the  following  questions: — 

a.  What  values  of  x  produce  a  positive  function  ? 

b.  For  what  values  of  x  is  the  function  negative  ? 

c.  If  x  =  — 1.5,  what  is  the  approximate  value 
of  /(*)? 


24  GRAPHIC  MA  THE  MA  TICS 

EXERCISES 

1.    Given  f{x)  =  x"  +  5  x  —7. 

(1)  Fill  in  the  values  omitted  in  the  following 
table:— 


Given  x  - 

Then  x*  = 

and  5x  = 
and 

=  x>  +  5  x  -7  = 

—3 

—2 

—1 

0 

1 

2 

3 

4 

5 

6 

7 

f(x) 

(2)  Plot  the  points  found  above,  and  draw  as 
steady  a  line  as  you  can  through  them. 

(3)  For  what  values  of  x  does  the  function  equal 
zero?    2?     3?     5?     10?    —6? 

(4)  For  what  values  of  x  is  the  function  negative  ? 

(5)  For  what  values  of  x  is  the  function  positive  ? 

(6)  When  x  =  — 2.5,  +  2.5  what  are  the  approxi- 
mate values  of  f(x)  ? 

(7)  How  many  times  does  the  graph  cut  the 
;r-axis  ? 

(8)  How  many  factors  has  the  expression 
x*  -f  5  x  —7  ? 

(9)*  What  are  they  approximately? 

(10)  Could  you  find  the  factors  exactly? 

(11)  If  you  were  to  plot  the  graph  of  f(x)  eeb-t2  + 
5  x  +  6  where  would  you  expect  it  to  cut  the 
.r-axis  ? 

2.     By   means  of   the   method   employed   in    the   last 
exercise,  plot  the  graph  of: — 

(1)  f{x)  =3j5  +  8^—4. 

(2)  f{x)  =4.r  2  —  8.r—  7. 


THE  FUNCTION  AND  THE  EQUA  TION  25 

(3)  f(x)  =  x*  +  3  x  +  1. 

(4)  f(x)  =  3  x  3  +  4  „r  3  —  8  *  —  7. 

3.  Draw   the  graph   of  the  parabola  f(x)  =  x*  using 
values  of  .r  between  +  and  —  5  inclusive. 

4.  Draw  the  graph  of  the  circle  f(x)  =  ±  V  36  —  x\ 
(Use  integral  values  of  x  between  ±  6  inclusive.) 


5.     Draw  the  graph  of  the  ellipse  f{x)  =  ±  ■£■ 1/  3  (4  —  .r2). 


6.  Draw  the  graph  of  the  hyperbola  f  (x)  s±  V  2x*  +  7. 

x* 

7.  Draw  the  graph  of  f(x)  = x  +  2. 

Those  of  us  who  know  the  trigonometric  ratios  can 
now  plot  the  graphs  of  functions  containing  them.  One 
example  will  be  sufficient  to  make  this  clear. 


Given  x  = 

0 

\ir  or  30° 

\tt  or  45° 

\tt  or  60° 

fix)  =  Sin  see 

0 

•5 

X_2_=707 

^  =.866 

i7r  or  90° 

\ir  or  120° 

|tt  or  135° 

77  or  180° 

fir  or  210° 

1 

43.866 

^or  .707 

0 

—.5 

|7T 

^7r 

3.77- 
2" 

fir 

ITT 

2  TT 

—.707 

—.866 

1 

—.866 

—.707 

0 

— .5 

—.707 

etc. 

—  4  TT 

-in 

etc. 

I                    -r 

s 

^eE„. 

-ZZ       5* 

Oil 

^v       ^ 

5r 

\ 

R 

~<M 

-+-        KU. 

Sr 

3: 

~r 

kU 

cor 

7 

J  1 

r 

,B|~ 

7 

^J!l 

.r 

>^ 

"R~ 

-p  0               -     - 

X 

r 

ig,«_ 

2 

r 

± 

-R  o 

w                                                      N^ 

I 

1 

i 

r 

f= 

1 " 

-H- 

26 


THE  FUNCTION  AND  THE  EQUA  TION  27 

It  is  readily  seen  that  /  (x)  =  sin  x  has  as  its  limiting 
values  +  1  and  — 1.  Therefore  we  shall  use  the  shorter 
axis  as  the  /(;r)-axis,  and  the  longer  one  as  the  x- 
axis.  On  the  .r-axis  the  unit  n  is  divided  into  sixths, 
fourths,  thirds,  and  halves,  therefore  we  shall  use  12 
divisions  to  the  unit  on  that  axis  (or  a  multiple  of  12). 
In  order  to  be  able  to  measure  tenths  on  the  /(;r)-axis 
we  shall  use  10  divisions  to  the  unit  on  that  axis.  Finally, 
so  that  the  graph  may  be  more  easily  drawn,  we  shall 
use  the  scale  24  to  it  on  the  .r-axis.  Plotting  the  points 
found  in  the  table  we  obtain  the  graph  shown  in  Fig.  6. 

Note. — Sin  x  is  an  example  of  what  is  called  a  periodic 
function — i.  e.,  a  function  which  repeats  the  same  values 
in  the  same  order  after  a  certain  period.  From  the  figure 
it  is  readily  seen  that  sin  (x  +  360°)  will  be  the  same 
as  sin  x.     Therefore  the  period  of  sin  x  is  360°  or  27T. 

EXERCISES 

1.  If  sin  x  =  .7  what  will  be  the  sine  of  (a)  (720°  +  x)? 

,(£)   (—360°  +x)l 

2.  Plot  the  graph  of  cos  x  =.f(x). 

3.  Plot  the  graph  of  f  (x)  =  tan  x. 

4.  Plot  the  graph  of/(x)  =  cot  x. 

5.  Plot  the  graph  of  f(x)  s  sec  x. 

6.  Plot  the  graph  of  f(x)  =  cosec  x. 

7.  Plot  the  graph  of  f(x)  =  sin  x  +  2. 

8.  Plot  the  graph  of  f  (x)  =  sin  x  +  cos  x. 

9.  Plot  the  graph  of  f(x)  =  sin  x  —  cos  x. 
10.  Plot  the  graph  of  f(x)  =  3  —  cos  x. 

ii.     Are  the  above  graphs  those  of  periodic  functions? 
If  so,  determine  the  period  of  each. 


28  GRAPHIC  MA  THEM  A  TICS 

THE  EQUATION 

From  what  has  been  said  in  the  beginning  of  this 
chapter  it  is  easily  seen  that  if  y  =  3  x  +  4,  x  would 
be  the  independent,  and  y  the  dependent  variable  and 
therefore  a  function  of  x.  If  then  we  call  our  axes  xx' 
and  yy'  in  place  of  ^r-axis  and  f(x) -axis,  we  may  plot  the 
graph  of  y  =  3  x  +  4  just  as  above  we  plotted  that  of 
f(x)  =  3  x  +  4. 

SINGLE  LINEAR  EQUATIONS 
EXERCISES 

1.  Draw  the  graph  of  y  =  5  x  — |,  and  from  it  find: 

(1)  The  value  of  x  when  y  =  0,  8,  10. 

(2)  The  value  of  y  when  x  =  2,  1,  — ^. 

2.  At  what  points  will  the  line  y  ==  4  x  +  6  cut  the 
axes  ?  What  is  the  easiest  way  to  find  these  points  ? 
What  then  is  a  simple  way  to  plot  an  equation  of  the 
first  degree  ?     (Such  equations  are  called  linear.)     Why? 

3.  Plot,  by  joining  the  points  where  the  line  cuts 
the  axes: 

(1)  y  =  x  f ;$.  (5)  x  =  —y  +  4. 

(2)  y  =  x  —  5.  (6)  5  x  +  2y  =  7. 

(3)  y  =  —  3  x  —  2.      (7)  9  ;r  +  7  J  —  8  =  0. 

(4)  y  =  —  3  x  +  2. 

4.  Can  you  plot  .r  =  — jK  by  the  method  suggested  in 
ex.  3  ?     Give  reason  for  your  answer. 

5.  Plot     (1)  x  =  —  y         (2)  x  =  5         (3)  y  =  —  8 

(4)  ^'=37  (5)^  =  |        (6).*-=^ 

6.  Give  the  equations  stating  that: 

(1)  A   point   is   always  10  units   from   a   given 
line  xx'. 

(2)  A  point  is  always  10  units  from  a  line  yy'. 

(3)  A  point  is  always  at  the  same  distance  from 
each  of  two  lines  which  intersect  at  right  angles. 


THE  FUNCTION  AND  THE  EQUATION  29 

SIMULTANEOUS  LINEAR  EQUATIONS 

7.  On  a  single  pair  of  axes  draw  the  graphs  of  the 
following  equations: 

(1)  3  x  +  4  7  =  18       (3)  \x  —  9  =  —  2y 

(2)  5  y  —  2  x  =  11       (4)  x  +  \y'm  12 

8.  From  the  graphs  in  Ex.  7  what  can  you  say  about 
equations  (1)  and  (2)  ?     (1)  and  (3)  ?     (1)  and  (4)  ? 

9.  Two  straight  lines  in  the  same  plane  in  general 
intersect  how  often?  May  they  do  otherwise?  Explain 
your  answer. 

10.  What  can  you  say  of  the  equations  of  two  straight 
lines  whose  graphs  intersect  once?  What  kind  of 
equations  must  they  be  to  give  such  result  ? 

The  line  or  group  of  lines  that  fulfills  a  given  condition 
is  termed  the  locus  of  that  condition.  For  instance,  the 
locus  of  the  condition  expressed  in  the  equation  x  =  3  is 
the  line  drawn  parallel  to  the  yy'  axis  at  a  distance  3  units 
to  the  right  of  it. 

Two  loci  are  said  to  be  coincident  when  every  point 
in  one  lies  on  a  corresponding  point  in  the  other,  or  in 
short,  when  they  have  all  points  in  common.  Two  loci 
are  said  to  be  parallel  when  they  have  no  point  in 
common,  and  they  are  said  to  intersect  when  they  have 
a  finite  number  of  points  in  common. 

11.  What  can  you  say  of  the  conditions  expressed  by 
(1)  and  (2),  Ex.  7  above?    by  (1)  and  (3)  ?    by  (1)  and  (4)? 

Two  equations  in  the  same  variables  are  said  to  be 
consistent  when  they  do  not  contradict  each  other,  and 
inconsistent  when  they  do. 

12.  Select  pairs  of  consistent  equations  from  Ex.  7. 

13.  Select  pairs  of  inconsistent  equations  from  Ex.  7, 


30 


GRAPHIC  MATHEMATICS 


14.  From  Ex.  7  can  you  tell  whether  all  consistent 
equations  can  be  solved  simultaneously?  Give  a  reason 
for  your  answer. 

15.  Do  you  suppose  that  inconsistent  equations  can  be 
solved  simultaneously  ? 

16.  How  was  equation  (3),  Ex.  7,  derived  from  equation 
(1)?  Are  they  consistent  then?  Would  you  say  they 
were  independent  of  each  other  ? 

17.  How  would  you  then  define  two  consistent  inde- 
pendent equations  ?  Select  two  such  equations*  from 
Ex.  7. 

18.  Arrange  answers  to  the  following  questions  just 
as  the  questions  are  arranged  and  underline  the  cor- 
responding words  and  phrases  in  the  two  columns. 


The  Linear  Equation 

1.  A  linear  equation  in 
two  variables  is  satisfied  by 
how  many  pairs  of  roots? 

2.  The  graph  of  a  linear 
equation  may  be  fixed  by 
how  many  pairs  of  its  roots? 

3.  In  general  two  linear 
equations  involving  the 
same  two  variables  have 
how  many  pairs  of  roots  in 
common? 

4.  May  two  linear  equa- 
tions in  the  same  two 
variables  have  more  than 
one  pair  of  roots  in  com- 
mon ?  What  kind  of  equa- 
tions are  they  then  ? 


The  Straight  Line 

1.  A  straight  line  con- 
tains how  many  points? 

2.  The  straight  line  is 
fixed  by  how  many  of  its 
points? 

3.  In  general  two  co- 
planar  straight  lines  have 
how  many  points  in  com- 
mon ? 

4.  May  two  coplanar 
straight  lines  have  more 
than  one  point  in  common  ? 

What  kinds  of  lines  are 
they? 


THE  FUNCTION  AND  THE  EQUA  TION  31 


5.  May  two  linear  equa- 
tions in  the  same  two 
variables  have  no  pair  of 
roots  in  common?  What 
kind  are  such  equations  ? 


5.  May  two  coplanar 
straight  lines  have  no  points 
in  common  ?  What  kind  of 
lines  are  they  ? 


19.     Solve  the  following  equations  graphically,  using 
a  new  pair  of  axes  for  the  solution  of  each  pair: 

(1)    \x—y  =  iZ>  (A\     j  ^  —  25  or  =  13 

K1)    \x+y=%  {*)     \y  +  62  =  50  x 

m    j*  +  »«'-2       ,«,     {5x  +  2y  =  8 
V)    \  y  =  %x       Is'     \2x  —  3y  =  —12 


(3)     j 


x  +  2y  =  7i 
2  x  +  y  =  7i 


SINGLE  QUADRATIC  EQUATION  AND  THOSE 
OF  HIGHER  DEGREE 

Suppose  we  were  now  asked  to  solve  the  equation 
x*  +  5  x  +  6  =  0.  Factoring,  we  see  at  a  glance  that 
(x  +  3)  (x  +  2)  =  0,  and  therefore  that  x  =  —  3  or  —  2. 

Let  us  now  see  how  we  might  have  found  these  values 
by  the  graphic  method.  From  what  we  have  learned  of 
functions  of  a  variable  and  of  the  single  linear  equation 
we  can  readily  plot  the  graph  of  y  =  x*  +  5  x  +  6. 
Here  we  are  not  interested,  however,  in  all  the  values  of  x, 
but  just  those  which  will  make  y  =  0.  Therefore,  having 
drawn  the  graph  of  f(x)  =  x*  +  5x-\-6  ovy~x2  +  5x  +  6, 
we  run  our  eye  along  it  until  we  find  the  points  at  which 
y  =  0,  or  in  short,  at  what   points   the  graph  cuts  the 


32 


GRAPHIC  MA  THEM  A  TICS 


,t'-axis.     At  these  points  we  find  the  values  of  x  to  be  —  2 
and  —  3  if  the  graph  is  accurately  drawn. 


2A 


(-'3,0) 


2,5-25)^ 

mffTtT 


(-2,0). 


/- 


■my 


Fig.7 

In   a   similar    manner    all    quadratic    equations — also 
those  of  higher  degree — may  be  solved. 

EXERCISES 

1.  Solve  graphically  the  equations: 

(1)  **  +  11  x  +  18  =  0. 

(2)  x*  —  7  x  +  12  =  0. 

(3)  2  x*  +  x  +  1  =  0. 

(4)  4  x*  +  4  x  +  1  =  0. 

(5)  x2  +  x  =  6. 

(6)  ;r2  +  3  =  G  x. 

(7)  9  .r2  —  5  x  —  2  =  0. 

(8)  .9  ,r2  —  4.68  x  =  —  4.36. 

(9)  3  x%  +  10  .r2  +  4.25  x  —  5  «-  0. 
(10)  ;r3  —  4.1  .r2  —  1.05  #  +  11.025  =  0. 

2.  How    many   times  does   the   locus  of  a  quadratic 
equation  in  x  cut  the  .r-axis? 


THE  FUNCTION  AND  THE  EQUA  TION  33 

3.  Show  graphically  the  character  of  the  roots  of  the 
equations: 

(1)  j2-3j-4  =  0. 

x~ 

(2)  '—  —  ;r  -f  2  =  0.     (Plot  using  values  between 

+  3  and  — 3.) 

(3)  x*  +  4  x  +  4  =  0. 

4.  How  does  the  graph  of  a  quadratic  equation  indi- 
cate the  fact  that  the  roots  of  the  equation  are: 

(1)  Real  and  unequal  ? 

(2)  Real  and  equal? 

(3)  Imaginary? 

SIMULTANEOUS    LINEAR    AND    QUADRATIC 
EQUATIONS 

Without  any  further  preparation  we  may  now  solve 
the  following  sets  of  simultaneous  equations. 

EXERCISES 

1.  In  what  points  does  the  straight  line  3  x  +  y  =  25 
cut  the  circle  x*  +  y2  =  65  ? 

2.  The  equation  of  a  circle  is  x*  +  y2  =  49,  and  the 
equation  of  a  chord  of  the  circle  is  13^+2j  =  49. 
Find  the  extremities  of  the  chord. 

3.  Solve  graphically  the  following  pairs  of  equationS{ 


(3)     Ux-iy       (j,-6)2  =  25 


4  x +  %  y  +  3  =  0 
Find  the  points  common  to  the  following  parabolas  and 
straight  lines: 

4.  y*  =  9  x,  3  x  +  30  =  7  y. 

5.  y  =  3  .f,  or  —  4  j  +  12  =  0. 

6.  y  =  4  4r,  *  =  6,  j  =  —  8,  x  =  0,  .r  =  —  4. 


34  GRAPHIC  MA  THEM  A  TICS 

7.  y2  =  Sxrx  +  y  =  6. 

8.  y2  —  4  #  —  8  j/  +  24  =  0,  3  y  —  2  x  ==  8. 

Find  the  points  of  intersection  of  the  following  ellipses 
and  straight  lines: 

9.  2  x2  +  3  j/2  =  14,  j  —  2  *  =  0. 

10.  2  ;r2  +  3  y2  =  35,  4  4f  +  9  j  =  35,  4  *  —  9  y  =  35. 

11.  9  ^2  +  64  J2  =  576,  2  J/  =  ;r  +  10,  2  J  =  .r  +  1. 

Find  the  points  common  to  the  following  hyperbolas 
and  straight  lines: 

12.  x2  —f  =  9,  4  #  +  5  y  =  40. 

13.  16  A'2  —  9j*  =  112,  9.r  +  16j  =  100,  16*  —  9y  =  28. 

SIMULTANEOUS  QUADRATIC  EQUATIONS 

Find  approximately  the  points  of  intersection  of  the 
following  loci: 

14.  2  x2  +  3  y2  =  14,  f  =  4:  x. 

15.  .r2  +  y2  =  10,  **  -f  7  j2  =  16. 

16.  x2  +  y2  =  25,  .rj/  =  5. 

MISCELLANEOUS  EXERCISES 

1.  Find  the  two  square  roots  of  6. 

(Hint:  Plot  the  graph  of  f  (x)  =  x\) 

2.  Find  the  three  cube  roots  of  8.    f(x)  =  x%. 

3.  Find  the  six  sixth  roots  of  1. 

4.  Which  of  the  above  roots  cannot  be  shown  graphi- 
cally ? 

5.  Write  the  equations  of  two  parallel  lines  and 
construct  them. 

6.  Write  the  general  equations  of  two  parallel  lines. 

7.  The  equation  of  the  circle  ax*  +  ay2  =  C  differs  in 
what  respect  from  the  equation  of  the  ellipse  ax2  +  by2  =  C? 
What  is  the  shape  of  the  ellipse  when  a  and  b  differ 


THE  FUNCTION  AND  THE  EQUTAION  35 

greatly   in  value?      When   a   and   b   are    nearly   equal? 
When  a  and  b  are  equal  ? 

8.     Draw  a  graph  by  means  of  which  American  money 
may  be  changed  to: — 

(1)  English  money.     (3)  French  money. 

(2)  German  money. 

7 


9.     Solve  graphically  j  ?£ff  *  {[ 


10.  Two  bodies  140  feet  apart  move  towards  each 
other,  the  first  at  the  rate  of  10  feet  per  second,  the 
second  four-fifths  as  fast.  How  long  before  they  are 
44  feet  apart  ? 


APPENDIX  TO  CHAPTER  II 

Draw  graphs  to  represent  the  statistics  given  in  the 
following  tables: 

1.     The  monthly  mean  maximum  temperature  Fahren- 
heit in  the  cities  noted  for  the  years  1872  to  1901: 


d 

J3 
ft 

Q. 
< 

a 

43 

•— > 

►—1 

< 

0 

0 

0 

26 

26 

32 

46 

5Q 

6q 

75 

73 

66 

53 

39 

35 

36 

42 

54 

66 

76 

81 

78 

71 

60 

49 

3i 

3i 

37 

50 

62 

72 

77 

76 

70 

58 

45 

3i 

33 

4i 

54 

64 

74 

80 

78 

72 

60 

45 

40 

43 

5i 

63 

74 

83 

87 

84 

78 

66 

52 

33 

35 

41 

54 

66 

76 

80 

78 

72 

61 

47 

74 

76 

77 

80 

84 

87 

89 

89 

87 

83 

78 

24 

28 

39 

57 

6o 

78 

83 

80 

71 

59 

41 

57 

6i 

67 

76 

84 

90 

02 

90 

86 

76 

66 

37 

3« 

44 

57 

68 

78 

82 

80 

74 

63 

51 

48 

5i 

56 

65 

75 

84 

88 

85 

79 

69 

59 

3i 

3i 

37 

50 

63 

73 

78 

76 

70 

57 

45 

20 

24 

36 

56 

68 

77 

83 

80 

7i 

57 

38 

Alpena,  Mich 

Boston,  Mass.  . . . 
Buffalo,  N.  Y.... 

Chicago,  111 

Cincinnati,  Ohio.. 
Cleveland,  Ohio.. 
Key  West,  Fla. . . 
La  Crosse,  Wis.. . 
Montgomery,  Ala 
New  York,  N.Y.. 

Norfolk,  Va 

Oswego,  N.  Y 

St.  Paul,  Minn... 


30 

36 
36 
43 
38 
74 
30 
58- 
41 

^6 

27 


2.     The  monthly  mean  minimum  temperature  Fahren- 
heit in  the  cities  noted  for  the  years  1872  to  1901: 


G 
at 

0) 
Pt4 

rt 

s 

< 

>> 

a 

C 

3 

>> 
"3 

bo 

< 

<L> 

in 

0 

0 

0 

12 

10 

16 

31 

41 

52 

57 

55 

49 

39 

28 

19 

20 

27 

37 

48 

^ 

63 

62 

55 

45 

34 

18 

17 

24 

35 

46 

58 

63 

61 

55 

44 

33 

16 

19 

27 

39 

4Q 

59 

65 

65 

58 

46 

32 

25 

27 

34 

45 

5? 

65 

69 

67 

60 

48 

37 

20 

20 

27 

38 

So 

59 

64 

62 

56 

45 

34 

65 

67 

68 

7i 

7S 

78 

79 

79 

78 

75 

7i 

7 

10 

22 

38 

So 

60 

64 

61 

53 

4i 

26 

39 

43 

48 

55 

63 

70 

73 

72 

67 

56 

46 

24 

24 

30 

40 

52 

61 

67 

66 

60 

48 

38 

33 

35 

39 

47 

58 

66 

7i 

70 

65 

54 

44 

17 

17 

24 

36 

46 

56 

62 

61 

54 

43 

33 

2 

7 

18 

36 

48 

58 

-62 

60 

5i 

39 

22 

Alpena,  Mich 

Boston,  Mass 

Buffalo,  N.Y 

Chicago,  111 

Cincinnati,  Ohio.. 
Cleveland,  Ohio.. 
Key  West,  Fla... 
La  Crosse,  Wis.. . 
Montgomery,  Ala 
New  York,  N.  Y.. 

Norfolk,  Va 

Oswego,  N.  Y. . . . 
St.  Paul, Minn.... 


19 
24 

24 

23 
30 

25 
66 

15 
40 
28 
36 
22 
11 


APPENDIX  TO  CHAPTER  II 


37 


The  preceding  material  as  well  as  what  follows  should 
be  made  use  of  in  various  ways  as  may  be  suggested  by 
both  pupils  and  teacher.  For  instance,  on  a  single  sheet 
of  cross-section  paper  make  a  diagram  showing  the  mean 
maximum  and  the  mean  minimum  temperatures  of 
Baltimore,  Md.,  using  a  dotted  line  to  show  the  mean 
maximum,  and  a  solid  line  to  show  the  mean  minimum. 
Using  red  ink  draw  a  line  showing  the  probable  mean 
temperature. 

3.     Average  amounts  of  precipitation  for  the  year  1904: 


San  Fran- 
cisco, Cal. 

Atlanta, 
Ga. 

4.75 

5-2 

3-31 

4 

02 

3-23 
i.8b 

.41 

5 
3 
3 

94 
69 
26 

.19 

.02 

4 
4 

03 
86 

.01 

4 

52 

•  44 
1.32 

3 
2 

55 
26 

2.70 

3 

44 

4.21 

4 

35 

Lincoln, 

Santa  Fe, 

Neb. 

New  Mex. 

.67 

.58 

.87 

•74 

I. 21 

•71 

2.67 

•75 

4-59 

1. 15 

4.36 

I.04 

4-13 

2.7 

3-39 

2-43 

2.14 

I.64 

2.07 

1.05 

•77 

.68 

.76 

.72 

Salt  Lake 
City,  Utah 


Yellow- 
stone Park, 
Wyo. 


Jan 

Feb 

March. . 
April... 

May 

June 

July.... 

Aug 

Sept 

Oct 

Nov 

Dec 


1-33 

1  4 

1.99 

2.13 

1.97 

•73 

•52 

•74 

.80 

1.5 
1-4 
1.43 


4 

92 

3 

23 

94 

65 

23 

07 

99 
09 

59 

86 


4.     The  population  of  New  York  City  to  the  nearest 
1,000  for  the  years  indicated: 


TEAR 

POPU- 
LATION 

YEAR 

POPU- 
LATION 

YEAR 

POPU- 
LATION 

1790 

1800 

I8IO 

1820 

33,000 

60,000 

96,000 

124,000 

I83O 

1840 

i8so 

i860 

203,000 
313,000 
5I6  000 
8o6,000 

I87O 

1880 

I89O 

1900. .... 

942,000 

1,206,000 
1,515,000 
3,437,000* 

*  All  Boroughs. 

Plot  the  above  correct  to  10,000  only. 


38 


APPENDIX  TO  CHAPTER  II 


5.     Immigration  into  the  United  States,  correct  to  the 
nearest  1,000: 


YEAR 

IMMI- 
GRANTS 

YEAR 

IMMI- 
GRANTS 

YEAR 

IMMI- 
GRANTS 

1820 

8,000 

i860 

133,000 

I89O 

455,000 

1825 

10,000 

1862 

72,000 

I892 

623,000 

1830 

23,000 

1865 

l8o,000 

1898 

229,000 

1835 

45,000 

1870 

387,000 

I9OO 

449,000 

I84O 

84,000 

,1875 

227,000 

1902 

649,000 

1845 

114,000 

!i88o 

457,000 

I903 

857,000 

I85O 

370,000 

1882 

789,000 

1904 

813,000 

1855 

201,000 

1885 

395,000 

6.     Income   and    Expenditures   of    the    United   States 
Government,  1876-1905.  (Record  to  the  nearest  $1,000,000) : 


YEAR 

REVENUE 

EXPENDITURES 

1876 

$287,482,039 
333,526,611 
323,690,706 
403,080,983 
313,390,075 
567,240,852 
543,423,859 

$258,459,797 
267,642,958 
260,226,935 
3l8,040,7H 
356,195,298 
487,7I^,7Q2 

1880 

1885 

I89O 

1895 

I9OO 

1905 

567,411,611 

7.     Public  Schools  in  the  United  States: 


1871. 
1876, 
1880, 
1885. 
1890, 


;2s 


W 


12 

3 

13 

7 

is 

1 

16 

7 

18 

5 

5.62 

6.06 

5-17 
6-6i 
7.60 


1895, 
1899 
1900, 
1905, 
1906 


e^£ 


20.4 
21 .9 
21.4 
23.4 
23.8 


•>*3 

5  0.2 JH 

<u  "a  S»S 

aas  Ow 
x  u  a 
W 


8.60 

9.13 
IO.04 
12.46 
12.94 


APPENDIX  TO  CHAPTER  II 


39 


8. 


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40 


APPENDIX  TO  CHAPTER  II 


9.     Density  of  population  per  square  mile,  of  States 
and  Territories,  1790-1900: 


e 


\4 


03 

C 

rtf 

0 

G 

M 

a 

a 

> 

O 

O 

^ 

>• 

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en 

a 

> 

a 

o 

0 

z 

z 

CL, 

7-1 

8.1 

97 

12.4 

9.8 

134 

20.1 

11.4 

18.0 

28.8 

13.2 

23-3 

40.3 

15.2 

30.0 

51.0 

15.5 

3»-3 

65.0 

17.9 

51.4 

81.5 

20.4 

64.6 

92.0 

22.1 

78.3 

106.7 

28.8 

QS.2 

1 26. 1 

33-3 

1 16.9 

152.6 

39-o 

1 40. 1 

1790 

1800, 
1810. 
1820, 
1830, 
1840, 
185a 
1 86a 
187a 
1880, 
1890, 
1900 


49-1 

51.8 

54.1 

56.8 

61.4 

64.0 

76.5 

95.0 

1 10.9 

128.5 

154.0 

187.5 


30.2 
32-8 
37.i 
37-1 
39-2 
39.8 
46.7 

57-3 
63.8 
74.8 
86.0 
943 


1.4 

2.8 

4.3 
5.8 
8.8 
11.7 
154 
17.9 
20.1 
26.1 
31.2 
37-6 


1.8 

5-5 
10.2 

14.1 

17.2 
19.5 
24.6 
28.9 
33-o 
41.2 
46.5 
537 


3-2 

5-1 

77 

1 0.0 

134 
16.8 
19.5 
21.0 
21.0 
21.7 
22.1 
23.2 


47.1 
52.6 
58.7 
65.1 

75.9 
91.8 
123.7 
I53.I 
181.3 
221.8 
278.5 
348.9 


15.8 
20.4 
23.8 
27.1 
29.9 
31.6 
35-3 
36.2 

35-3 
38.5 
41.8 

45-7 


634 

637 

70.9 

76.6 

89.6 

100.3 

136.0 

160.9 

200.3 

254.9 

318.4 

407.0 


10.     Native  and  Foreign   born  population   of   various 
cities,  correct  to  the  nearest  100  : 


CITY 

\ 

1870 

1880 

1890 

1900 

Washington,  D.  C: 

Native  born 

95,400 

I33.IOO 

211,600 

258,600 

Foreign  born 

13,800 

1 4,200 

18,800 

20,100 

Buffalo,  N.  Y.: 

Native  born 

71,500 

103,900 

166,200 

248,100 

Foreign  born 

46,200 

51,300 

89,500 

104,300 

San  Francisco,  Cal.: 

Native  born 

75,800 

129,800 

172,200 

225,900 

Foreign  born 

73,800 

104,200 

126,800 

116,900 

Portland,  Oreg.: 

Native  born 

5,700 

11,300 

29,100 

64,600 

Foreign  born 

2,600 

6,300 

17,300 

25,900 

Atlanta,  Ga.: 

Native  born 

20,700 

36,000 

63,700 

87,300 

Foreign  born 

1,100 

1,400 

1,900 

2,500 

Savannah,  Ga  : 

Native  born 

24,600 

27,700 

39,800 

50,800 

Foreign  born 

3700 

3.000 

3,4oo 

3400 

Hoboken,  N.  J.: 

Native  born 

10,000 

18,000 

26,300 

38,000 

Foreign  born 

10,300 

13,000 

17,400 

21,400 

APPENDIX  TO  CHAPTER  II 


41 


11.     The  population  of  a  few  States,  by  color  at  each 
census: 


MAINE 

South  Carolina 

Georgia 

White 

Colored 

White 

Colored 

White 

Colored 

1790 

96,002 

538 

140,178 

108,895 

52,886 

29,662 

1800 

150,901 

818 

196,255 

149,336 

102,261 

60,425 

1810 

227,736 

969 

214,196 

200,919 

I454I4 

107,019 

1820 

297,406 

929 

237440 

265,301 

189,570 

15^419 

1830 

398,263 

1,192 

257,863 

323,322 

296,806 

220,017 

1840 

500,438 

i>355 

259,084 

335,3H 

407,695 

283,697 

1850 

581,813 

i,356 

274,563 

393,944 

521,572 

384,613 

i860 

626,952 

1,327 

291,388 

412,320 

591,588 

465,698 

1870 

625,309 

1,606 

289,792 

415,814 

638,967 

545,U2 

1880 

647,485 

1,451 

391,245 

604,332 

817,047 

725J33 

1890 

659,896 

1,190 

462,215 

688,934 

978,538 

858,815 

1900 

693»H7 

1,319 

557,995 

782,321 

1,181,518 

1,034,813 

12.     The  areas  of   Indian  Reservations  for  the  years 
indicated  given  in  square  miles: 


YEAR 

ARIZONA 

IOWA 

NEBRASKA 

N.  CAROLINA 

1880 

I89O.   ... 

I9OO 

1907 

4,832.5 
IO,3l7.5 
23,673 
26,532.7 

I 

2 

4-5 
4.63 

682 
214 
Il6 

23.08 

102 
102 

153.5 
98.77 

13.  Departures  of  passengers  from  seaports  of  the 
United  States  for  foreign  countries  1868  to  1907,  correct 
to  the  nearest  100: 


YEAR 

TOTAL 

YEAR 

TOTAL 

YEAR 

TOTAL 

1868..... 

32,SOO 

1879 

51400 

I898 

94,600 

I87O 

33,600 

I885 

87,800 

I9OO 

155,900 

1872 

39,900 

I89O 

105,900 

1905 

201,200 

1873 

52,lOO 

189I 

I07,IOO 

1907 

224,900 

1876 

46,400 

1893 

95,100 

I878 

55,200 

1894 

1 2 1 ,900 

42 


APPENDIX  TO  CHAPTER'II 


14.     Records  of  Cereal  Crops,  1866  to  1907 


Wheat— Average 


Per  acre 


^    RS   O 


Oats— Average 


Per  acre 


Barley— Average 


Per  acre 


Bushels 

[i.6 

[2.1 

13.6 

12.4 

[1.6 
[1.9 
12.7 
12.3 

0.5 
3-9 

f{ 

3-1 

0.2 

1.6 
3.0 
04 
2.4 
2.1 
1.1 
2.9 
1.1 
5.3 

34 
14 
3-2 
37 
2.4 
34 
5.3 
2.3 
2.3 
5.0 

4-5 
2.9 

2.5 
45 
5-5 
4.0 


Dollars 

16.83 

13.17 
10.38 

11.73 
13-24 
1335 
13.56 

10.65 
9.91 

10.09 

14.65 
10.15 

15.27 
12.48 

12.12 

12.02 

IO.52 

8.38 

8.05 

8.54 

8.25 

IO.32 

8.98 

9.28 

12.86 

8-35 
6.16 
6.48 
6.99 

8.97 

10.86 

8.92 

7.17 
7.61 

9-37 

9.14 

8.96 

11.58 

1083 

10.37 
12.26 


Bushels 
30.2 
25.9 
26.4 

30.5 
28.1 
30.6 
30.2 

277 
22.1 

297 
24.O 

31.7 
31-4 
28.7 
25.8 

247 
26.4 
28.1 

274 
27.6 
26.4 

254 
26.0 
27.4 
19.8 
28.9 
244 
23-4 
24.5 
29.6 

257 

27.2 

28.4 
30.2 
29.6 
25.8 

28.4 
32.1 
34.0 
31.2 
237 


Dollars 
I0.6I 

11-53 
II.OO 
II.58 
IO.97 
II.07 

9-03 

9-59 

10.38 

9.52 

777 
9.01 

772 

9.50 

9.28 

11.48 

9-89 
9.20 
7.58 
7.88 

7.87 
774 
7.24 
6.26 
8.40 
9.08 

773 
6.88 

7.95 
5.87 
4.81 

575 
7.23 
7.52 

7.63 
10.29 
10.60 

9.68 
10.05 

9.88 

9.89 
10.51 


Bushels 
22.9 
22  7 
24.4 
27.9 
237 
24.O 
I9.2 
23.I 
20.6 
20.6 

21.9 

21.3 

23.6 
24.0 

24.5 

20.9 

21.5 

21. 1 

23-5 
21.4 
22.4 
I9.6 
21.3 

24.3 
21.4 
25.9 
23.6 
21.7 
194 
26.4 
23.6 

24.5 
21.6 

25.5 

20.4 
25.6 
29.0 
26.4 
27.2 

26.8 
28.3 
23.8 


APPENDIX  TO  CHAPTER  II 


43 


15.  Value  of  gold  and  silver  produced  in  the  United 
States.  (Plot  correct  to  the  half-million  dollars,  showing 
on  separate  sheets  the  gold  and  silver  production,  and  on 
one  sheet  the  amount  of  gold  produced  in  California, 
other  States  and  Territories,  and  the  total  amount  pro- 
duced.) 


i860 
1861 
1862 
1863 
1864 

1865 
1866 
1867 
1868 
1869 

1870 
1871 
1872 
1873 
1874 

1875 
1876 
1877 
1878 
1879 

1880 
1881 
1882 
1883 
1884 

1885 
1886 
1887 
1888 


California 


Dollars 
45,000,000 
40,000,000 
34,700,000 
30,000,000 
26,600,000 

28,500,000 
25,500,000 
25,000,000 
22,000,000 
22,500,000 

25,000,000 
20,000,000 
19,000,000 
17,000,000 
17,500,000 

17,617,000 
17,000,000 
15,000,000 
15,300,000 
16,000,000 

17,500,000 
l8,2OO,0OO 
16,800,000 
14,120,000 
13,600,000 

12,700,000 
14,725,000 
13,400,000 
12,750,000 
13,000,000 


Other  States 
and  Territories 


Dollars 
1,000,000 
3,000,000 
4,500,000 
1 0,000,000 
19,500,000 

24,725,000 
28,000,000 
26,725,000 
26,000,000 
27,000,000 

25,000,000 
23,500,000 
1 7,000,000 
19,000,000 
15,990,900 

15,850,900 
22,929,200 
31,897,400 
35,906,400 
22,900,000 

18,500.000 
16,500,000 
15,700,000 
15,880,000 
17,200,000 

19,101,000 
20,144,000 
19,736,000 
20,417,500 
19,967,000 


Total 


Dollars 
46,000,000 
43,000,000 
39,200,000 
40,000,000 
46,100,000 

53,225,000 
53,500,000 
51,725,000 
48,000,000 
49,500,000 

50,000,000 
43,500,000 
36,000,000 
36,000,000 
33,490,900 

33,467,900 
39,Q29,200 
46,897,400 
51,206,400 
38,900,000 

36,000,000 
34,700,000 
32,500,000 
30,000,000 
30,800,000 

31,801,000 
34,869,000 
33,136,000 
33,167,500 
32,967,000 


Dollars 

156,800 
2,o62,000 
4,684,800 
8,842,300 

11,443,000 

11,642,200 
10,356,400 
13,866,200 
12,306,900 
12,297,600 

16,434,000 
23,588,300 
29,396,400 
35,881,600 
36,917,500 

30,485,900 
34,919,800 
36,991,500 
40,401,000 

35,477.100 

34,717,000 
37,657,500 
41,105,900 
39,618,400 
41,921,300 

42,503,500 
39,482,400 
40,887,200 
43,045,100 
46,838,400 


44 


APPENDIX  TO  CHAPTER  II 


15.    Value  of  Gold  and  Silver  Produced  in  the 
United  States — Continued. 


GOLD 

YEAR 

California 

Other  States 
and  Territories 

Total 

SILVER 

1 89O 

Dollars 
12,500,000 
12,600,000 
12,000,000 
12,080,000 
13.570,000 

14,929,000 
15,235,900 
I4,6l8,300 
15,637,900 
15,197,800 
I5,8l6,200 

Dollars 
20,345,000 
20,575,000 
21,015,000 
23,875,000 
25,930,000 

3I,68l,000 
37,852,400 
42,744,700 
48,825,100 
55,855,600 
63,354,800 

Dollars 
32,845,000 
33,]75,000 
33,015,000 
35,955,000 
39,500,000 

46,6lO,000 
53,088,000 
57.363.000 
64,463,000 
71,053,400 
79,171,000 

Dollars 
57,242,100 
57,630,000 
55,662,500 
46,800,000 
31,422,100 

36,445,500 
39,654,600 
32,316,000 
32,Il8,400 
32,859,000 
35,741,140 

189I 

1892 

I893 

I894 

1895 

I896 

I897 

1898 

I899 

IQOO 

16.  Anthracite  and  bituminous  coal  production  in  the 
United  States.  (Show  record  on  a  single  pair  of  axes 
and  correct  to  one  million.) 


YEAR 

Total 

Total 

Total 

Total 

Anthracite 

Bituminous 

Anthracite 

Bituminous 

Tons 

Tons 

Tons 

Tons 

1880 

25,580,180 

38,242,641 

I9OI 

60,302,264 

201,572,572 

189O 

41,489,858 

99,377.073 

I902 

37,024,582 

232,252,596 

1897 

47,036,389 

131,739,681 

1903 

66,678,392 

252,389,837 

I898 

47,705.125 

148,702,257 

1904 

65,382,842 

248,738,941 

1899 

54,030,536 

172,524,099 

J905 

69,405,958 

281,239,252 

I900 

51,309,214 

189,480,097 

17.  Number  of  employees  thrown  out  of  work  because 
of  strikes.  Correct  to  nearest  hundred.  (Plot  correct  to 
1,000.) 


YEAR 

NUMBER 

YEAR 

NUMBER 

l88l 

129,500 

I89O 

352,000 

1882 

154,700 

I89I 

299,OCO 

1883 

149,800 

I892 

206,700 

1884 

147,100 

1893 

265,QCO 

I885 

242,700 

1894 

660,400 

1886 

508,000 

I89S 

302,400 

I887 

379,700 

I896 

241,200 

1888 

147,700 

1897 

408,400 

I889 

249,6C0       i 

I898 

249,000 

1899 

1900 
1901 
1902 

1903 
1904 
1905 


417,100 
505,100 

543.400 

659,800 
656,100 
517,200 
221,700 


APPENDIX  TO  CHAPTER  II 
18.     Number  of  strikes: 


45 


CALENDAR 
YEAR 

Ordered  by 
labor  organ- 
izations 

Not  ordered 
by  labor  or- 
ganizations 

CALENDAR 
YEAR 

Ordered  by 
labor  organ- 
izations 

Not  ordered 
by  labor  or- 
ganizations 

l88l 

1882 

1883 

I884 

1885 

1886 

I887 

1888 

I889 

189O 

189I 

I892 

1893 

223 
220 

271 
240 

357 
763 
952 
616 
724 
1,306 
1,284 
918 
906 

248 

234 
207 
203 
288 
669 

483 
288 

351 
525 
432 
380 
399 

I894 

1895 

I896 

1897 

1898 

1899 

1 900 

I9OI 

1902 

1903 

1904 

|1905 

847 

658 

662 

596 

638 

1,115 

1,164 

2,218 

2,474 

2,754 

1,895 

1.552 

501 

555 
363 
482 
418 
682 

6ll 

706 

688 
740 
412 

525 

19.     Number   of    Post   Offices    in  the    United   States, 
correct  to  the  nearest  500. 


YEAR  ENDED 
JUNE  30TH 


879 
880 
88l 
882 
883 
884 
885 

886 
887 
888 
889 
890 
891, 
892 
893 


POST  OFFICES 


41,000 

43,000 

44,500 

46,000 
48,000 
50,000 

51*500 

53,500 

55,000 

57,500 

59,000 
62,500 
64,500 
67,000 
68,500 


YEAR  ENDED 
JUNE  30TH 


I894 
1895 
1896 
I897 
I898 

I899 
I9OO 
1 901 
1902 
1903 
1904 
1905 
1 906 
1907 


POST  OFFICES 


70,000 
70,000 
70,500 
71,000 
73,500 
75,000 
76,500 
77,000 
76,000 

74,000 
71,000 
68,000 
65,500 


20.     Number  of  offices  of  the  Postal  Telegraph  Cable 
Company,  correct  to  the  nearest  100. 


46 


APPENDIX  TO  CHAPTER  II 


YEAR 

OFFICES 

YEAR 

OFFICES 

YEAR 

OFFICES 

YEAR 

OFFICES 

I885.. 

300 

189I.. 

1,200 

I897.. 

9,900 

1903.. 

20,000 

1886.. 

400 

I892. . 

I,400 

1898.. 

II,IOO 

I9O4.. 

2I,IOO 

T887.. 

600 

1993- • 

1,600 

I899.. 

12,700 

1905.. 

23,100 

1888.. 

700 

1894.. 

1,800 

1900. . 

13,100 

I906. . 

25,300 

I889.. 

800 

1895.. 

2,100 

1901. . 

14,900 

1907.. 

25,500 

I89O.. 

1,000 

1896.. 

9,IOO 

I 902 . . 

l6,200 

21.  Table  showing  the  increase  in  mileage  of  railroad 
in  operation  in  the  United  States.  Given  correct  to 
nearest  unit: 


-a 
a 

"be 

03 

a 

"3 

'a 

3 

■d5« 

Hi 

a 

Oi 

GRAND 

YEAR 

W 

<U-w 

as  0 

— 

a  m;=: 

£ 

£ 

0 

TOTAL 

£ 

I2 

Z< 

~i> 

J3 

£t 

S 

£ 

s 

<u 
V 

0 

O 

O 

O 

i860.. 

3,660 

6,353 

9,583 

5,463 

3,727 

1,102 

655 

23 

30,626 

1870.. 

4494 

io,577 

H,70I 

6,481 

5,106 

4,625 

5,004 

i,934 

52,922 

1880.. 

5,982 

15, '47 

25,109 

8,474 

6,995 

14,085 

12,347 

5,128 

93,267 

I89O. . 

6,832 

20,038 

36,976 

17,301 

13,343 

32,888 

27,294 

12,031 

166,703 

1900. . 

7,501 

22,385 

41,138 

21,905 

(6,211 

37,530 

32,106 

15,486 

194,262 

I9O4.. 

7,619 

23.150 

43,252 

23,589 

18,297 

44,852 

34,307 

17,328 

212,394 

I9O5.. 

7,681 

23,408 

43,959 

24,180 

19,026 

46,06l 

35,157 

17,869 

217,341 

I906. . 

7,729 

23,559 

44,427 

24,897 

19,735 

47,447 

36,097 

18,743 

222,634 

22.     Average   receipts   per   ton   per   mile   on    leading 
railroads  of  the  United  States: 


YEAR 

CENTS 

YEAR 

CENTS 

I87O 

1880 

I89O 

1900 

1902 

4.50 
2.21 
1.50 

•93 
I.OI 

1903 

I904 

1905 

I9O6. 

.98 

•99 
•94 
•93 

APPENDIX  TO  CHAPTER  II 


47 


23.     Number  of  persons  killed  by  railway  accidents  in 
the  United  States,  1888  to  1906: 


YEAR  ENDED 
JUNE  30TH 

. EMPLOYEES 

PASSENGERS 

OTHER  PERSONS 

1888        

2,070 
1,972 

2,451 
2,660 

2,554 

2,727 
1,823 
1,811 
1,861 
1,693 

1,958 
2,210 
2,550 
2,675 
2,669 

3,606 
3,632 
3,36i 
3,929 

315 
3IO 
286 
293 
376 

299 

324 
170 
l8l 
222 

221 
239 
249 
282 

345 

355 
441 

537 
359 

2,897 
3,541 
3,598 
4,076 
4,217 

4,320 
4,300 

4,155 
4,406 

l889 

I  89O 

I  89I 

1892 

l8o^ 

I  894 

i8oq 

1896 

1897 

4,522 
4,680 

1898 

1  8qq 

4,674 
5,066 
5,498 
5.274 

5.879 

5.973 
5,805 
6,330 

1900 

1901 

1902 

IQO^ 

I9O4    

I9O5 

I906 

24.  Table  showing  the  number  of  sailing  and  steam 
vessels  in  use  in  the  United  States,  correct  to  nearest 
100: 


YEAR  ENDED 
JUNE  30TH 

SAILING  VESSELS 

STEAM  VESSELS 

1879 

20,600 
l8,700 
17,700 
17,100 
15,900 
l6,500 
1 4,900 

4,600 
5,400 
5,900 
6,500 
6,800 

1884 

1889 

I894 

I899 

1902 

7,700 
10,100 

1907..  < 

48 


APPENDIX  TO  CHAPTER  II 


25.     Comparison  of   the   number  of   various   kinds  of 
vessels  built  in  the  United  States,  1881-1907: 


SAILING  VESSELS 

STEAM  VESSELS 

CO 

YEAR  ENDED 

-a 

co 

"5 

0 

^H 

D 

JUNE  30TH 

to  ■_, 

CO 

a 

0 

d, 

% 

^  rt 

CO 

TOTAL 

B.Cd 

So 

0 

X! 

O 

O 

£* 

0. 
0 

as 

c5 

m 

PQ 

Ifl 

w 

in 

w 

p- 

U 

PQ 

l88l 

29 

3 

318 

143 

ss 

105 

284 

57 

114 

1,108 

I883 

33 

2 

567 

119 

46 

90 

303 

42 

66 

1,268 

1884 

24 

2 

533 

147 

32 

103 

27S 

33 

41 

1,190 

I885 

11 

379 

143 

39 

86 

213 

21 

28 

920 

1886 

8 

I 

276 

120 

18 

80 

142 

23 

47 

715 

I887 

7 

I 

258 

l8l 

24 

69 

206 

36 

62 

844 

1888 

4 

27s 

144 

33 

84 

313 

40 

121 

1,014 

1889 

1 

296 

192 

28 

87 

32S 

88 

60 

1,077 

I89O 

10 

347 

148 

26 

99 

28  s 

40 

96 

1,051 

I89I 

13 

I 

447 

272 

28 

ill 

349 

57 

106 

1,384 

1892 

8 

423 

4'5 

26 

[©S 

307 

37 

74 

1,395 

1893 

8 

I 

303 

181 

19 

9^ 

268- 

28 

55 

956 

I894 

3 

253 

221 

26 

61 

206 

14 

54 

838 

1895 

1 

188 

208 

17 

70 

161 

11 

38 

694 

1896 

2 

215 

152 

25 

84 

177 

13 

55 

723 

1897 

1 

160 

177 

20 

88 

180 

70 

195 

891 

I898 

1 

159 

199 

15 

170 

209 

20 

179 

952 

1899 

3 

.... 

223 

194 

14 

182 

243 

13 

401 

1,273 

I9OO 

4 

281 

219 

19 

117 

286 

38 

483 

1,447 

1901 

6 



259 

261 

21 

131 

354 

79 

469 

1,580 

1902 

9 

316 

256 

27 

137 

415 

44 

287 

1,491 

I903 

3 

298 

169 

28 

131 

392 

19 

271 

i,3H 

1904 

203 

127 

13 

161 

439 

25 

216 

1,184 

1905 

I9S 

US 

10 

164 

386 

30 

202 

I,IC2 

I906 

154 

75 

16 

147 

487 

83 

259 

1,221 

I907 

81 

66 

15 

149 

510 

62 

274 

1,157 

26.     Lives  lost  through  disasters  to  vessels  on  rivers  of 
the  United  States: 


YEAR 

LIVES  LOST 

YEAR 

LIVES  LOST 

YEAR 

LIVES  LOST 

1887 

1888 

I889 

189O 

I89I 

1892 

1893 

89 
17 
78 
63 

129 

50 

34 

I894 

1895 

I896 

1897 

1898 

1899 

I9OO 

29 
15 

50 

7 
25 
4i 
18 

I9OI 

1902 

1903 

1904 

1905 

I906 

1907 

19 

157 

35 
30 
20 
34 
24 

APPENDIX  TO  CHAPTER  II 


49 


27.     Table  showing  some  work  performed  by  Revenue 
Cutter  Service. 


19OI 

1902 

1903 

1904 

1905 

1906 

1907 

Lives  saved  (actually  res- 
cued) from  drowning 

Persons  in  distress  taken 
on  board  and  cared  for. . 

178 
IOI 
107 
178 

55 
538 

IOI 

191 

19 

3i 

7i 

230 

24 

47 

154 

494 

18 

187 
521 
262 

17 
1,285 

131 

378 

4i 

78 

138 

Vessels  seized  or  reported 
for  violation  of  law 

319 

28.  Table  showing  total  amount  of  merchandise  im- 
ported into  and  exported  from  the  United  States.  (Correct 
to  the  nearest  million): 


ended 
June  30th 

TOTAL  VALUE 

TOTAL  VALUE 

ended 
June  30th 

TOTAL  VALUE 

TOTAL  VALUE 

IMPORTS 

EXPORTS 

IMPORTS 

EXPORTS 

Million 

Million 

Million 

Million 

Dollars 

Dollars 

Dollars 

Dollars 

1870.... 

436 

377 

1889.... 

745 

730 

1871.... 

520 

428 

189O 

789 

845 

1872.... 

627 

428 

1891.... 

845 

872 

1873.... 

642 

505 

1892 

827 

I,Ol6 

1874.... 

567 

569 

1893-.. 

866 

831 

1875.... 

533 

499 

1894.... 

655 

869 

1876.... 

461 

526 

1895.... 

732 

793 

1877-..  • 

45i 

590 

1896.... 

780 

863 

1878.... 

437 

681 

1897.... 

765 

1,032 

1879.... 

446 

698 

1898.... 

616 

1,210 

1880.... 

668 

824 

1899.... 

697 

1,204 

l88l.... 

643 

884    * 

I9OO 

850 

i,37i 

1882.... 

725 

783 

I90I 

823 

1,460 

1883.... 

723 

804 

I902 

903 

1,355 

1884.... 

668 

725 

I903-... 

1,026 

1,392 

1885.... 

578 

727 

I904.... 

99 » 

i,435 

1886.... 

635 

•  666 

I905.... 

1,118 

1,492 

1887.... 

692 

703 

I906 

1,227 

1,718 

1888.... 

724 

684 

1907.... 

i,434 

1,854 

50 


APPENDIX  TO  CHAPTER  II 


29.     Table  showing  value  of  exports  of  cotton  goods  of 
domestic  manufacture.     (Correct  to  nearest  million): 


YEAR 

MILLION 
DOLLARS 

YEAR 

MILLION 
DOLLARS 

YEAR 

MILLION 
DOLLARS 

I856.... 

7 

l874... 

3 

l892 

13 

1857..: 

6 

1875.... 

4 

1893.... 

12 

I858.... 

6 

I876.... 

8 

I894.... 

14 

1859.... 

8 

1877.... 

10 

1895.... 

H 

i860.... 

11 

1878.... 

11 

I896.... 

17 

l86l.... 

8 

1879.... 

11 

I897-.. 

21 

l862.... 

3 

l880.... 

10 

I898.... 

17 

I863.... 

3 

l88l.... 

H 

I899.... 

24 

I864.... 

1 

1882.... 

13 

1900 

24 

I865.... 

3 

I883.... 

13 

1901 

20 

1866.... 

2 

I884.... 

12 

1902 

32 

I867.... 

5 

1885.... 

12 

1903.... 

32 

1868.... 

5 

1886.... 

14 

1904.... 

22 

I869.... 

6 

I887.... 

15 

1905.... 

50 

I87O.... 

4 

1888.... 

13 

I906 

53 

I87I.... 

4 

I889.... 

10 

1907.... 

32 

1872.... 

2 

I89O.... 

10 

1873.... 

3 

I89I.... 

14 

30.     Annual  average  price  in  dollars  per  ton  of  coal: 


YEAR 

ANTHRACITE 

BITUMINOUS 

YEAR 

ANTHRACITE 

BITUMINOUS 

I850.... 

3-64 

I87O.... 

4.39 

472 

1853.... 

370 

3  30 

I875-.. 

4-39 

4-35 

I855.... 

4-49 

3.89^ 

1877.... 

•         2.59 

3.15 

i860   .  .  . 

340 

3-49 

jl88o.... 

4-53 

3-75 

l86l.... 

3-39 

344 

I885.... 

4.10 

2.25 

1862.... 

4.14 

4.23 

I  89O 

3.92^ 

2.60 

I863.... 

6.06 

5-57 

1895.... 

3-50 

2.00 

1864.... 

8-39 

6.84 

1898.... 

3.50 

1.60 

1865.... 

7.86 

7-57 

I9OO...  . 

347 

2.50 

1866.... 

5.80 

5-94 

1905.... 

4.50 

2.60 

APPENDIX  TO  CHAPTER  II 


51 


31.     Value  of  sugar  and  molasses   imported  into  the 
United  States.     (To  the  nearest  half  million): 


Year 

Year 

ended 

SUGAR 

MOLASSES 

ended 

SUGAR 

MOLASSES 

June  30th 

June  30th 

Dollars 

Dollars 

Dollars 

Dollars 

in  Millions 

in  Millions 

in  Millions 

in  Millions 

l86l.... 

30.5 

4.0 

1885.... 

72.5 

4.0 

1862.... 

20.5 

3-5 

1886.... 

81.0 

5-5 

1863.... 

I9.O 

4-5 

1887.... 

78.5 

5-5 

1864. .  .  . 

29.5 

7-5 

1888.... 

74.0 

5-5 

1865.... 

27.5 

7.5 

1889.... 

88.5 

5.0 

1866.... 

40.5 

7-5 

189O 

96.O 

5.0 

1867.... 

36.O 

11. 5 

189I.... 

106.0 

2.5 

1868.... 

49-5 

12.0 

1892 

104.5 

3-o 

1869.... 

60.5 

12.0 

1893-.. 

1 16.5 

2.0 

1870.... 

57.0 

13.0 

1894.... 

127.O 

2.0 

1871.... 

64.5 

10.0 

1895.... 

76.5 

1.5 

1872.... 

8l.O 

10.5 

1896... 

89.O 

•5 

I873.... 

82.5 

10. 0 

1897.... 

99.O 

•5 

1874.... 

82.0 

II.O 

1898.... 

60.5 

•5 

1875.... 

73-5 

11.5 

1899.... 

95.O 

1.0 

1876.... 

58.O 

8.0 

I9OO 

IOI.O 

1.0 

1877.... 

85.O 

8.0 

I90I 

9O.5 

1.0 

1878.... 

73-o 

7.0 

I902 

55.0 

1.0 

1879.... 

72.0 

7.0 

I903.... 

72.0 

1.0 

1880.... 

80.0 

8.5 

I904.... 

72.0 

1.0 

I88l.... 

86.5 

6.5 

I905.... 

97-5 

1.0 

1882.... 

90.5 

1 0.0 

I906 

85.5 

.5 

1883.... 

91.5 

7-5 

I907.... 

93-0 

1.0 

1884.... 

98.0 

5-5 

32.     Average  food  cost  per  workingman's  family  in  the 
United  States,  1 890-1 906: 


United  States, 

United  States, 

United  States, 

2,567  families 

YEAR 

2,567  families 

2,567  families 

Dollars 

Dollars 

Dollars 

1890 

318.20 

I896.... 

296.76 

1I902 

344-61 

1891 

322.55 

I897.... 

299.24 

I903.... 

342.75 

1892 

316.65 

I898.... 

306  70 

I904.... 

347-10 

1893.... 

32441 

I899-.. 

309.19 

I905.... 

349-27 

1894.... 

309.81 

I9OO 

314.16 

I906 

359-53 

1895.... 

303.91 

1901 

326.90 

52 


APPENDIX  TO  CHAPTER  II 


33.     Relative  wholesale  prices  of  raw  and  manufactured 
commodities  in  the  United  States,  1890-1906: 


Raw  Com- 

Manufactured 

Raw  Com- 

Manufactured 

YEAR 

modities 

Commodities 

YEAR 

modities 

Commodities 

I89O 

II5.0 

II2.3 

I899.... 

I05.9 

IOO.7 

189I 

1 16.3 

1 10.6 

I9OO.  .  . 

III.9 

1 10.2 

I892 

IO7.9 

105.6 

I9OI 

III.4 

107.8 

I893.... 

IO4.4 

105.9 

1902 

122.4 

1 10.6 

1894.... 

93-2 

96.8 

I9O3.... 

122.7 

111.5 

1895.... 

91.7 

94.0 

1904.... 

1 197 

111,3 

I896.... 

84.O 

91.9 

1905.... 

121. 2 

1 14.6 

1897.... 

87.6 

90.1 

I906 

I25.9 

121.6 

I898.... 

94.O 

93-3 

34.     Amount  of  money  in  circulation  per  capita  in  the 
United  States,  1884-1907: 


Money  in  cir- 

Money in  cir- 

Money in  cir- 

YEAR 

culation  per 

YEAR 

culation  per 

YEAR 

culation  per 

capita 

capita 

capita 

Dollars 

Dollars 

Dollars 

I884.... 

22.65 

I892.... 

24.56 

I9OO 

26.94 

I885.... 

23.02 

1893.... 

24.03 

I9OI 

27.98 

1886.... 

21.82 

1894.... 

24.52 

I902 

28.43 

I887.... 

22.45 

I895.... 

23.20 

1903.... 

29.42 

1888.... 

22.88 

I896.... 

21.41 

I9O4.... 

30.77 

I889.... 

22.52 

1897.... 

22.87 

1905.... 

31.08 

I89O.... 

22.82 

I898.... 

25.15 

I906 

32.32 

I89I 

23.42 

I899.... 

25.58 

1907.... 

32.22 

35.     Receipts  and  expenditures  per  capita  in  the  United 
States: 


YEAR 

Receipts 

Expenditures 

YEAR 

Receipts 

Expenditures 

I898.... 

$6.77 

$7.29 

1993.... 

$8.59 

$7,920 

I899..      . 

8.21 

9.41 

1904.... 

8.36 

8.868 

1900 

8.78 

7-73 

I905.... 

8.37 

8.649 

1901..  .  . 

8.99 

7-994 

1906 

9.OI 

8.702 

1902 

8.65 

7.496 

1907.... 

9.84 

8.859 

APPENDIX  TO  CHAPTER  II 


53 


36.     Debt  per  capita  less  cash  in  the  Treasury  of  the 
United  States: 


882 


888 


Debt  per  cap. 

less  cash  in 

YEAR 

Treas. 

Dollars 

3546 

I89O 

31.91 

I89I.     .. 

28.66 

1892 

26.20 

1893.... 

24.50 

1894.... 

22.34 

1895.... 

20.03 

I896.... 

1772 

I897.... 

15.92 

I898.... 

Debt  per  cap. 

less  cash  in 

YEAR 

Treas. 



Dollars 

14.22 

1899.... 

13-34 

1900 

12.93 

I9OI 

12.64 

I902 

13-30 

I903.... 

13.08 

1904.... 

13.60 

1905.... 

1378 

1906 

I4.08 

1907.... 

Debt  per  cap. 

less  cash  In 

Treas. 


Dollars 
15.55 
14.52 

1345 
12.27 
II.5I 
H.83 
II.9I 
II.46 
I0.22 


37.     Tables  showing  progress  of  the  United  States: 


YEAR 

<u  S 

Q.    . 

I 

a 
0 

"3. 

»-  cti 

0J.~  O 

«-  a-d 

•<=  ft 

CO 

In 

1 

92 

8    g" 

■b       rt 

S--  0 

a 

J*     £ 

w 

1800 

I8I0 

1820 

I83O 

I84O 

1850 

1855./:.. 
l860./.... 

1865 

1870.     ... 

1875 

l880.... 

1885 

189O 

1895 

I9OO 

1905 

I906 

1907 

6.4I 
3.62 
4.68 
6.25 
829 

778 
9.14 
IO.39 
II.48 
I2.74 

14.51 

I6.57 
18.55' 
20.69 

22.77 

25.I4 

27.38 
27.82 

28.35 

307.69 
5I3-93 
779.83 

850.20 

1,038.57 
1,117.01 

I,l64.79 

15.63. 
7-34 
942 

377 

.21 

2.74 

I.3I 

I.9I 

76.98 

60.46 

47.53 
38.27 
24.50 
14.22 
13-08 

14.52 
1 1.9 1 
11.45 
I0.22 

5.00 

7-59 

6.94 

6.79 

1 0.9 1 

12.02 
15.34 
13.85 
20.57 

17.50 

17.16 
19.41 

23.02 
22.82 
23.20 

26.94 
31.08 

32.32 
32.22 

17  19 

11.80 

7.72 

4.87 

5.76 

748 
9.46 

11.25 
6.87 

11.06 

11.97 
12.51 
10.32 

12.35 
10.61 

10.88 
13.08 

14.41 
16.55 

1337 

9.22 
7.22 

5.57 
7.25 

6.23 

8.03 
10.61 

4.78 

9-77 

11.36 
16.43 
12.94 
13.50 
11.51 

17.96 
17-94 
20.40 
21.60 

54 


APPENDIX  TO  CHAPTER  II 


38.     Number  of  newspapers  and  periodicals  published 
in  the  United  States: 


YEAR 

NUMBER 

YEAR 

184O.. 
185O.. 
i860.. 
1870.. 

NUMBER 

YEAR 

NUMBER 

YEAR 

NUMBER 

I  800 .  . 
I8l0.. 
1820. . 
183O.. 

359 
861 

1,403 

2,526 
4,051 
5,78l 

1875.. 
1880.. 
1885.. 
I89O. . 

7,870 
9,723 
13494 
16,948 

1995.. 
1900. . 
1905.. 
1907.. 

20,395 
20,806 
23,146 
21,735 

39.     The  number  of  students  in  colleges,  universities, 
and  schools  of  technology  in  the  United  States: 


YEAR 

NUMBER 

YEAR 

NUMBER 

1875 i    32,175 

1880 1     38,227 

1885 42,573 

1891 58,405 

I896 

IQOO 

I903 

86,864 
98,923 
108,381 

40.     The  number  of  volumes   in    all   libraries    in   the 
United  States: 


YEAR 

NUMBER 

(per  100  inhabitants) 

YEAR 

NUMBER 

(per  100  inhabitants) 

1875 

1885 

I89I 

26 

35 
4i 

I896 

IQOO 

1903 

47 
59 
68 

41.  According  to  the  "  Revista  Scientifico-Industriale  " 
the  cost  of  sugar  at  London  and  Paris  from  the  middle  of 
the  13th  centurv  was  as  follows: 


YEAR 

LONDON 

PARIS 

YEAR 

LONDON 

PARIS 

1260..^  . 

$187 

1542.... 

$.62 

I3OO.... 

2.27 

I550.... 

$.83 

1350.... 

1. 51 

1598.... 

•97 

1372.... 

$5 

•17 

1600 

72 

1400 

2.10 

I65O 

72 

1426...  . 

2 

.62 

I  700 

.48 

1450.... 

2.72 

1750.... 

.19 

1482 

2 

•50 

1800 

•34 

I500 

.48 

MATHEMATICS 


Plane  Geometry  with  Problems  and  Applications 

By  H.  E.  Slaught,  Associate  Professor  of  Mathematics  in  the  Uni- 
versity of  Chicago,  and  N.  J.  LENNES,  Instructor  in  Mathematics  in 
Columbia  University,  New  York  City.  i2mo,  cloth,  288  pages.  Price, 
$1.00. 

THE  two  main  objects  that  the  authors  have  had  in  view  in 
preparing  this  book  are  to  develop  in  the  pupil,  gradually 
and  imperceptibly,  the  power  and  the  habit  of  deductive  reasoning, 
and  to  teach  the  pupils  to  recognize  the  essential  facts  of  elemen- 
tary geometry  as  properties  of  the  space  in  which  they  live, 
-  not  merely  as  statements  in  a  book. 

These  two  objects  the  book  seeks  to  accomplish  in  the  follow- 
ing ways: — 

1.  The  simplification  of  the  first  five  chapters  by  the  exclusion 
of  some  theorems  found  in  current  books. 

2.  By  introducing  many  applications  of  special  interest  to 
pupils,  and  by  including  only  such  concrete  problems  as  fairly 
come  within  the  knowledge  of  the  average  pupil.  Such  problems 
may  be  found  in  tile  patterns,  parquet  floors,  linoleums,  wall 
papers,  steel  ceilings,  grill  work,  ornamental  windows,  etc.,  and 
they  furnish  a  large  variety  of  simple  exercises  both  for  geometric 
construction  and  proofs  and  for  algebraic  computation. 

3.  The  pupil  approaches  the  formal  logic  of  geometry  by 
natural  and  gradual  processes.  At  the  outset  the  treatment  is 
informal ;  the  more  formal  development  that  follows  guides  the 
student  by  questions,  outlines,  and  other  devices,  into  an  attitude 
of  mental  independence  and  an  appreciation  of  clear  reasoning. 

The  arrangement  of  the  text  is  adapted  to  three  grades  of 
courses :  — 

(a)  A  minimum  course,  providing  as  much  material  as  is  found 
in  the  briefest  books  now  in  use. 

(b)  A  medium  course,  fully  covering  the  entrance  require- 
ments of  any  college  or  technical  school. 

(c)  An  extended  course,  furnishing  ample  work  for  those 
schools  where  the  students  are  more  mature,  or  where  more  time 
can  be  given  to  the  subject. 

67 


MATHEMATICS 


High   School  Algebra:    Elementary  Course 

By  H.  E.  Slaught,  Associate  Professor  of  Mathematics  in  the  Univer- 
sity of  Chicago,  and  N.  J.  Lennes,  Instructor  in  Mathematics  in 
Columbia  University,  New  York  City.  i2mo,  cloth,  309  pages. 
Price,  $1.00. 

THIS  book  embodies  the  methods  of  what  might  be  called  the 
new  school  of  Algebra  teaching,  as  revealed  in  numerous 
recent  discussions  and  articles. 

Some  of  the  important  features  of  the  Elementary  Course  are  :  — 

I.  Algebra  is  vitally  and  persistently  connected  with  Arithme- 
tic. Each  principle  in  the  book  is  first  studied  in  its  application 
to  numbers  in  the  Arabic  notation.  Principles  of  Algebra  are 
thus  connected  with  those  already  known  in  Arithmetic. 

II.  The  principles  of  Algebra  are  enunciated  in  a  small  number 
of  easy  statements,  eighteen  in  all.  The  purpose  of  these  prin- 
ciples is  to  arrange  in  simple  form  a  codification  of  those  opera- 
tions of  Algebra  that  are  sufficiently  different  from  the  ordinary 
operations  of  elementary  Arithmetic  to  require  special  emphasis. 

III.  The  main  purpose  of  the  book  is  the  solution  of  problems 
rather  than  the  construction  of  a  purely  theoretical  doctrine  as  an 
end  in  itself.  An  attempt  is  made  to  connect  each  principle  in  a 
vital  manner  with  the  learner's  experience  by  using  it  in  the  solu- 
tion of  a  large  number  of  simple  problems. 

IV.  The  descriptive  problems,  as  distinguished  from  the  mere 
examples,  are  over  seven  hundred  in  number,  about  twice  as 
many  as  are  contained  in  any  other  book.  A  large  share  of  these 
problems  deal  with  simple  geometrical  and  physical  data.  Many 
others  contain  real  data,  so  that  the  pupil  feels  that  he  is  deal- 
ing with  problems  connected  with  life.  The  authors  believe 
that  it  is  possible  to  make  problems  so  interesting  as  to  answer 
the  ordinary  schoolboy  query  —  "What  is  Algebra  good  for?1' 

V.  The  order  of  topics  has  been  changed  from  the  traditional 
one,  so  that  those  subjects  will  come  first  which  have  a  direct 
bearing  on  the  solution  of  problems,  which  is  looked  upon  as  the 
first  purpose  of  Algebra. 

68 


Mathematics 


High   School  Algebra:    Advanced  Course 

By  H.  E.  SLAUGHT,  of  the  University  of  Chicago,  and  N.  J.  LENNES, 
of  Columbia  University,  New  York  City.  i2mo,  cloth,  202  pages. 
Price,  65  cents. 

THE  Advanced  Course  contains  a  complete  review  of  the  sub- 
jects treated  in  the  Elementary  Course,  together  with  enough 
additional  topics  to  meet  the  requirements  of  the  most  exacting 
technical  and  scientific  schools. 

The  subjects  already  familiar  from  use  of  the  Elementary  Course 
are  treated  from  a  more  mature  standpoint.  The  theorems  are 
framed  from  a  definite  body  of  axioms.  The  main  purposes  of 
this  part  of  the  book  are :  (a)  To  study  the  logical  aspects  of 
elementary  Algebra ;  (b)  to  give  the  needed  drill  in  performing 
algebraic  operations.  To  this  end  the  exercises  are  considerably 
more  complicated  than  in  the  Elementary  Course. 

The  chapters  containing  material  beyond  the  scope  of  the  Ele- 
mentary Course  are  worked  out  in  greater  detail.  The  chapters 
on  Quadratics  and  Radicals  contain  rich  sets  of  examples  based 
largely  upon  a  few  geometrical  relations  which  the  pupils  used 
freely  in  the  grammar  schools.  The  chapters  of  Ratio,  Propor- 
tion, and  Progression  also  contain  extensive  sets  of  problems. 

High  School  Algebra:    Complete  Course 

By  H.  E.  SLAUGHT,  of  the  University  of  Chicago,  and  N.J.  LENNES, 
of  Columbia  University,  New  York  City.  i2mo,  cloth,  509  pages. 
Price,  $1.20. 

THIS  book  is  the  Elementary  Course  and  the  Advanced  Course 
bound  in  one  volume,  and  is  designed  to  furnish  enough 
algebra  to  fit  the  entrance  requirements  of  any  college  or  uni- 
versity. 


A  Primary  Algebra 


By  J.  W.  MacDonald,  Agent  of  the  Massachusetts  Board  of  Educa- 
tion.    i6mo,  cloth.    The  Student's  Manual,  92  pages.     Price,  30  cents. 
The  Complete  Edition,  for  teachers,  218  pages.     Price,  75  cents. 
69 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 

REC'D  LD 

FhB  27  1957 

ooQtf*9101 

Hfttf  i  n  &&# 

4Nov'6nEW 

REC'D  LD 

0CT21I960 

'■■>     •-        ■    ■:-■■    ■■      ■'      - 

'  •    V  — '     .     :  .:    -,■ 

RECD  LD  DEC 

9  '69  -3PM 

LD  21-100m-6,'56                                  TT   .Gen.eral  ¥£**?     . 
(B9311S10 )  476                                       UmveM^crf  Cdif orma 

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